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Okay, so let's begin. We're going to continue with our readmuller boards. So let me remind you once again how we are thinking about these boards. So we have, first, if you want to think of readmuller boards, after readmuller boards, the length to parent, you're going to think of binary loader boards. How many of them do you have? 312VM. And then this will be a polynomial of a total degree. I'll simply say degree by degree. I mean total degree. That's when I go up. And then what do you do? Go from here to a code word. And the thing about this code word, it basically has an evaluation of X below 100. So you go through every single vector in 01M, evaluate this polynomial F from those vectors. You'll get how many points. So how many gates here? There are 2 power m vectors. So if you evaluate it and make a vector, you'll have that 2 power m gates. So perfectly, that's from 0 to 0. So this was one view of the code word. Another thing to say is that this is actually a Boolean function. So there's one more view. Then you can also think of the matrix ideas and all that.  One type of property is that you need to readmuller code. So if you want to readmuller code, opt-order readmuller code of length 2 power m, you can do a uv plus v constriction. You will come from the r-th order readmuller code of length 2 power m minus 1. And we will come from the r minus 1-th order readmuller code of length 2 power m minus 1. So that's the rest idea. So you will come from belongs to Rn, r comma r minus 1. We will belong to Rn, r minus 1 comma r minus 1. So from here you can build up the generated matrix in a nice way if you like. That's one view of this one. So from this property, we showed the readmull equals 2 power m minus r. So that's another property which is quite nice to show. The thing we're going to show today is few more interesting properties. And once again, we will play with this motion of the polynomial evaluation and the vector interpretation. So both of them will go hand in hand. So you think of a polynomial and then the vector that results when you evaluate it on every single possible value for the variables, you get a vector and then you play with that vector. So this relationship is what gives all the nice properties for readmuller code. So that's where everything comes from. So we'll continue with that. And the property that we see today is about the Boolean. Boolean of r comma m. So you can show that if you take the r power readmuller code of r, Boolean will be another readmuller code of r minus r minus 1. So this is the result. So the first thing we're checking is that the number of vectors that we expect in the Boolean is the same as under I can say. So here, the dimension is what?   So that's the dimension.  Are these two the same? Well, it has to be the same. I gave you the result. But how will you show that these two are the same? 2 power m is the same. Okay. So you can do that.  You write it that a, subtract it out. All the first things go away. And then you use the identity that m choose k is the same as m choose m minus k. We'll get the results. It's not very hard to do that. It's straightforward computation. So you see, at least as far as dimension is concerned, these two match. So if I show that the r of the chess is contained in the m of chess, I'll be done. If dimension is the same, both of them have the same number of elements. So if I show that the right-hand side, that is the codewords of the m minus r minus 1 power rate miller code, the codewords of those is contained in the dual of the r m rate miller code in the left-hand side. What do I mean by saying a codeword here is contained in the dual? It should be up to the level 2. Every single codeword of the original two.   Once I show that, I'm done. So that's what we will show. We will show the following.  So let's say it's a codeword. And a b which belongs to r m r comma m. That will be half. We have what? a dot b equal to what? 0. This is what I want to show. So now there are various ways of doing this. So of course we are going to use the polynomial idea. Remember the vector a is associated with the polynomial of degree 1. m minus r minus 1. The vector b is associated with the polynomial of degree r. So at most r. So what will happen if I multiply these two polynomials? I can multiply these two polynomials. What will the evaluation of that product polynomial correspond to? How will it be related to a and b? Do you see that? So the question I'm asking is a will be a 0, a 1 over to a 2 power m minus 1. If a i is what? Some polynomial f of b 1 will be given. You are looking at a vector i. So what do you think of i? So what is i? i is just an integer from 0 to 2 power m minus 1. So I can now take with m bit representation. So every integer from 0 to 2 power m minus 1 has an m bit representation. I'll take its m bit representation, take the first bit and use it for v1. Second bit I'll use it for v2. So instead of thinking of substituting variables as bits, m bits, I can think of it as an integer from 0 to 2 power m minus 1. Nothing wrong with it. So i is an integer from 0 to 2 power m minus 1. It has an m bit representation. How many bits are there in an m bit representation? m bits. First bit, second bit, third bit, all the way to m bit. I will simply use v1 equal to the first bit, v2 equal to the second bit, v3 equal to the third bit. So that is what I mean by this substitution. So this i, I can think of as a value that is taken by v1 through vn. Is it clear? Is it clear as a way to give out a very clear explanation for this? Is it clear? So I can think of, suppose m is 4, I can think of the fourth bit between the four bit vectors that's going all the way to the front like this. Or equivalently, what can I think of it as? 0, 1, 2, all the way to sustain. So if I say I substitute, let's say some 9, what does it mean? I'll actually be substituting 1001. So both of them are equal. I can think of it as 4. This is my vector a. Likewise, how do you know that the degree of excess less than or equal to m minus r minus 1? Likewise, I'll have b between 0 and b1. I have to go to b to power minus 1. And where is bi? Some g evaluated at the same i. That's where it goes. So how do I know the degree of g? Degree of g is less than or equal to r. Now what happens when I multiply s and g? I'll get some other polynomial. And its degree will be less than or equal to m minus 1. Now if I evaluate that product at i, what will I get? I'll get a i times b i. So what happens if I evaluate f of b1, b2 bm times b of b1, b2 bm at i? What will I get? a i times b i. Is that okay? So this new vector, let's call it some b, that I'm going to define. I'll define a new vector b, which is b0 b1 b2 bar minus 1. And this bi will basically be the product evaluated at i. And you know this bi equals a i b i. So what happens with our b summation bi bar 2? So if I show that the weight of b is going to be even all the time, then I'm done. Am I right? All I have to show is the weight of b is zero. So this will be zero is the weight of b is even. How do I know the weight of b will be even? What is the degree of this guy? What's going to be equal to m minus 1? So what happens if I take a polynomial of degree less than or equal to m minus 1 and evaluate it at all the possibilities? Will I get an even weight vector? I will get an even weight. So that implies b belongs to the regular code of m minus 1 m. And that clearly implies weight of b is even. So it's a very simple proof except that you go back and forth between this vector and polynomial and it can be very consistent. The first time you see it will never really be clear. But if you think about it, it's a very kind of an obvious easy proof. Okay? Is it okay? So I'm too slip correct. All right, so you take a vector from the first code. You see that it's an evaluation of a polynomial at all points. Take a vector from another code. It's an evaluation of a polynomial at every point. And if I take the two corresponding coordinates a and d and multiply together it's the same as evaluating the product polynomial at the same position. But interestingly, the product polynomial has degree less than or equal to m minus 1 which means it belongs to the regular code m minus 1, m. And that you know is an even weight code, isn't it? The vector in that code is even weight. And that shows you that there's two after them. So this will work. So even m minus r minus 2. So if this degree is less than or equal to m minus r minus 2 also the same logic will work. So why can I not conclude that the regular code of m minus r minus 2 is the dual? The dimension will not agree. The dimension will be smaller. There are more codes in the dual.  So it means that this is the dual. There's nothing more. So you can stop there. That doesn't, that also makes sense, right?  So that this also does happen. It's not a very surprising fact. And all these codes are basically contained in each other. So it has to work that way. Okay. So this makes a very interesting family. So if you think about cyclic codes, right? So we had first of all, you know, of course there are very easy samples to make. So if you look at linear codes, the dual of a linear code is also a linear code. So it's a nice, close step for it has. Likewise for cyclic codes interestingly, the dual is also cyclic. And for read Miller codes again, what do you get? Dual is read Miller code. Okay. So all these things are nice. I mean from VCH codes also you can show dual is VCH. Read Solomon codes also you can show dual is the read Solomon extracts. All those things you can show. So it's nice how do you say it? It's nice close property with respect to the dual. And that maybe you can see it's a product. Okay. So even if you don't have any immediate use for it, you'll see that there's a use for it. Okay. So here's the sooner enough that it's good to know that the dual also has some good properties. In the decoding, we can use this code. All right. Any questions? On how I did this? So the first time you see it, this proof can be confusing. That's how all proofs in read Miller codes go. We'll use the property of the code and the polynomial evaluation and then go back and forth. Okay. Is that all right? So let's take a simple example. For much more than equals three, look at what happens. Okay. So it's nice to see the one equal to three example. So I'm looking at three. So let's write down the vectors. So one is going to be one, one, one, one, one, one. The three is going to be zero, zero, zero, zero, one, one, one. The two is going to be zero, zero, one, one, one. The one is going to be zero, one, zero, one. Zero, one, zero, one. We're going to have to write it back with three, two. This will be zero, zero, zero, one. The three, one. This will be zero, zero, zero, zero, zero, one. Then it will be zero, zero, zero, zero, one. So, if I want to generate a matrix for rate mill upon, well the 0th order is really not that interesting, but anyway let us do it. 0, 3, it could be what? Only the first row, so you get the repetition, the correct, get the repetition form. So, it will be n equals 8 repetition form. And then you have the first order rate mill upon, which will be an 844 code. What will be the gender matrix? The first 4 rows. And then if I want the second order rate mill upon, it is going to be 874 code. So, this will be the even rate code. So, let me not talk about the r of 3, 3, which is identity code, not really. So, you can use our result now and see that, so the 0, 3, what should be its doing? m minus r minus 1. So, you put r equal to 0 and this 3 you get 2, 1. So, what happens if I put r equal to 1? You get the same thing at length. So, this guy is in fact a self-dual code. The code is its own dual. So, it is an interesting little matrix there. So, let us look at the gender matrix for that guy, this is the gender matrix. So, it has a very nice symmetric looking structure, so self-dual. So, one property of the self-dual code is this k will be equal to n by 2. That is correct. So, the self-dual k equals 1 minus k, right? So, k should be equal to n by 2. So, what the satisfied work? And another way to check the self-dual properties, any two rows of the generator matrix will be orthogonal to each other. And that is kind of enough to show self-dual. So, you look at any two rows of the generator matrix here, they will all be orthogonal to each other. Because g is equal to h, g and h are the same, you know, g times h transpose has to be 0. So, g times g transpose has to be 0. In both conditions you check, you will see the self-dual, you can do that very easily. And you can do if m equals 4, you will get a more, I mean this, the list of binary vectors in 16, it is a bit more painful to write it down. You want to write it down. To do that, you will get a different list. So, what I am going to do is just write down for m equals 4, 4 is 1, 2 rows, 2 rows again, m of r m, 3 rows of m, and 3, 4 b, 2 rows of each other. So, this will be the replication code and this will be the even-made code. And what will you have? r m of 1, 4 and k will have 2, 4. So, these two will be 2 rows of each other. So, what is the dimension here? 6 k comma 5 and 8, 8 is the minimum distance. So, what is the dimension here? It has to be 11. The minimum distance is 4. Is that right? So, there is another useful and simple rule you can keep in mind if you are worried about how do I select the vectors. So, if you have 16 vectors, right? There are 16 basis vectors, 1, b 1, b 2, b 3, b 4, so on and on the way to the last day, b 1, b 2, b 3, b 4. If I want the basis vectors for r m 2 comma 4, another rule to keep in mind. So, I may not have taught it like that. It is very easy also to think about it. You look at the 16 basis vectors, you select all the vectors whose weight is greater than or equal to 4. In this list, we select all the vectors whose weight is greater than or equal to 4. They will be the top 11. It is not very difficult to think about it. You can see that once it goes to b 1, b 2, b 3, your weight will go down to 2. Only if you have 2 of them, your weight will be greater than or equal to 4. So, that is a useful rule to keep in mind. Among the 16 vectors, all the vectors whose weight is greater than or equal to 4 make the generator matrix or the 16, 11, 4. The minimum distance plays a nice role that way. So, in this construction, it is not very useful because anyway you can go from the top and pick it up that in the other chronicle product construction I gave you, I said there is a fabrication of this which shows up. There you may not know where which is coming. There you can use this rule very nicely. If you want the app that I read more about, what will I do? I look at all the rows of that matrix whose weight is greater than or equal to 2 power m minus r and that will give me my generator matrix. So, that is a rule that is useful to remember. So, just push the side information. So, these two are the ones. So, let us just do anything aside. We have once again r m of 0, 4, 0, 5, r m of 4, 5, you have r m of 2, 5, r m of 3, 5 and then you have r m of 2, 5. What will happen to 2, 5? It will be self-dual. So, you can show every time m is odd, r m of m minus 1 by 2, m will be a self-dual. And you can do quickly dimensions if you like because these things may be different. Then first thing 32, 6 out of the 16. What is this one? 32, 26. What? 4, right? This thing is what? 32, should be 16. You can do the addition that should be 16 and then you have 8. So, it is a nice order to these numbers. What is the nice order to these numbers? 2 power 5, 2 power 4, 2 power 3. So, that is a nice thing to keep in mind. So, in fact, that will be generally true.  What will be its coordinates? 2 power m, 2 power m, 2 power m, 2 power m minus 1. Why should it be 2 power m minus 1? Because I know it is self-dual. So, it has to be equal to that. And then what will be the thing? 2 power m minus m plus 1 by 2, right? So, this will be the parameter. So, in this nice case, it turns out that m plus 1 by 2 is the same as m minus 2. It may not happen in general. So, 2 power m has to be odd. But m minus m minus 1 by 2, that will be m plus 1. So, this will be a self-dual code. Okay. So, that is all I wanted to say about Bruins and the structure of this product. So, let us move on to encoding. We will talk about encoding. So, here we will use one more interesting property. So, the first encoding we can do is to use the generator matrix. So, whether likely to be problems if we use the generator matrix. So, you might say it could be a huge matrix. You have to store something. But we have the recursive structure, right? So, maybe you do not store too much. I mean, maybe you can get away with some savings there. But there is one more problem which you cannot go over to use the generator matrix again. What is that problem? It is not systematic. So, it is also being not systematic. And maybe you do not like it too much. Okay. So, it turns out we can use one more structure for the readmuller codes. It turns out readmuller codes are very, very closely related to cyclic codes. Okay. So, in fact, you can puncture one bit of the readmuller code and get a cyclic code. So, that is the fact. Okay. So, basically, another idea is to use the puncturing. So, you puncture, puncture one bit of the readmuller code to get a cyclic code. Okay. So, how do you do that? I will tell you how to do this. I will prove this very rigorously. I will prove this too painful and unnecessary. So, I will show you how it is done. And you will see it is kind of obvious just to work out that way. But the puncture, this is not an interesting idea. So, how do you use this idea? Suppose I tell you that this is true. Suppose I tell you that the code which you can puncture in the readmuller code. So, once you puncture it, you get a cyclic code. Then what you can do? You encode the cyclic code even systematically. Then how do you generate the punctured code again? You add an overall parity. Okay. So, puncturing is some parity of the original thing. So, you can do that. So, this can be used very nicely to build an encoder. So, this property can be used to build a systematic encoder. Okay. That is the idea. So, which one should we puncture? That is the first question. So, I will tell you the answer. And then we will do this first by example. Okay. So, we will do a very simple example first. And then we will go through and see why it should be true in general. Okay. So, for example, I am going to take this. So, we will begin with the simplest example. We will take Rm1, R3. Okay. We will take the first order readmuller code of length 8. So, how do we render the matrix? What should we do now? I will take the dual of this code. Okay. So, what is the dual of this code? Oh, Rm1, R3 is a dual. So, it does not matter. Okay. So, that is nice. So, let us take the dual. So, apparently, it makes it because also the generator matrix is what? You have 1111 on top. I am going to write the 1111 at the bottom because that is what we are maybe used to. Okay. I am going to write the remaining stuff. Okay. So, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1. Okay. Let this remind you of something. That is the form of the tidal digit matrix. Now, suddenly, I know it is the tidal digit matrix also. Obviously, I did not know that. Now, I know this is also the tidal digit matrix. But when you start it, it takes away. You should be reminded of something. What are you reminded of? The Hamming code. Right? The 7-4 Hamming code. Are you reminded of that? Which part of it reminds you of the 7-4 Hamming code? This girl. Okay. This part is just the 7-4. 3. Nothing. Right? So, what is the additional operation that has been done? We saw this before. Yeah. We are adding, extending the code. We are extending to add an overall tidal digit matrix. So, extension and puncturing are opposites of each other. Right? So, if you take the extended version and puncture the last bit, what will you get? You should get... No, I should be careful there. Extension and sharpening are opposites of each other. If you extend and puncture, what will happen? If I puncture this, what will happen? Okay. Oh, no. What happens when I puncture it? It should... So, puncturing is easier than the extended matrix. The bariatric matrix puncturing is a bit more painful. So, what should we do when we puncture? Yeah, you are right.  This will also go away and that will also go away. So, we will get back to the 7-4-3 Hamming code. What do I know about the 7-4-3 Hamming code? It is just a cyclic code. Not in this order. In what order is it a cyclic code? How should I order the columns to get a cyclic code in the Hamming code? Yeah, take the integral element of GF8 and do 1 alpha, alpha squared, alpha 3, all the way to alpha power 6. Not in this order. Don't do 0, 0, 0, 1, 0, 1, 0. First one will be 0, 0, 0, 1. Next will be alpha. It should be 0, 1, 0. Next will be alpha squared. It should be 1, 0, 0. Next will be alpha 3, which is 0, 1, 1, etc. So, you put it in a different order. You get a cyclic code. Okay. So, I will show the regular code and then evaluate the polynomial in a different order. Okay. Right? What do you get when you do that? You get a Hamming code, which is a cyclic code. So, that's the idea. Okay. Essentially that is the idea. So, this, so this guy is cyclic after all column notation. Okay. So, the same thing you can show is true for an arbitrary readmuller code. So, there it's a little bit more difficult to prove that present rate. So, it's not as easy as to read for extended the code. I can easily do this for any problem, but the extension maybe is not too hard. Okay. So, I have to go to the DL of a readmuller code. It should be again a readmuller code. It will have an extension like this. So, I can function it. Then I will have the evaluations at non-zero vectors. I have to re-order the evaluation according to the primitive element. Okay. So, I can always take the primitive element in GF2 param and re-order the evaluation. So, first I will be just evaluating in the natural bit model. I have to now re-order it in terms of the alpha, alpha square analogy. Come in some crazy order. Once you re-order it and evaluate it, it turns out that readmuller code is cyclic. So, that's the idea. Okay. So, let me write it down. I will write down the result. I will give you a hint as to how you prove it, if you won't really prove it. Okay. So, this is the result. The main result is you take a readmuller code. So, in general, you take a readmuller code of R command. Okay. The puncture, first bit. So, what do I mean by first bit? Puncture the evaluation corresponding to? It's quite easy to do this. So, what do I mean by puncture the evaluation corresponding to 0000? Simply then evaluate it. Okay. So, you start only with the non-zero vectors for v1 through vm. Okay. And then what do I do? Evaluate in the order 1, alpha, alpha square, algebra 2, alpha, part 2, param, minus 2. And what is alpha? It's a primitive element of gf2 param. What do I mean by evaluating a polynomial in m binary variables at alpha? What do I mean by that? What does that mean? How can I evaluate f of v1, v2, vm at alpha, part i? What does that mean? Previously, we saw evaluating it at the integer i, where i was between 0 and 2 param, minus 1. So, now what do you do for alpha, part i? Well, I take the vector representation. I know it is n bits. So, you put v1 equal to the first bit in the vector representation. v2 equal to the second bit in the vector representation. Okay. Except that now, in the previously when I did simple binary representation, it will come in the natural binary order. Now, it won't come in the natural binary order. It will come in some crazy order according to the primitive sequence in the value of the arithmetic. Okay. So, that's the only thing. So, you evaluate that. The only thing I skip this, I am not evaluating it at 0. Okay. 0 I am skipping. I am only evaluating it once from alpha to part i, minus 2. So, if you do that, it turns out this readmiller code will be the punctured readmiller code. Okay. So, in fact, after puncturing, we usually do a star. Okay. So, it will be not the puncturing. So, rm of r comma n star is the punctured version after the first person puncture. And then evaluated in this sequence, you get a cyclic code. Okay. So, this becomes cyclic. Okay. So, all right. Do you have a question? Is it okay? That's the idea. I mean, I am not... It's just a notation. You see, rm of r comma n is the code of the punctured. If I put a star, it means that I am puncturing. Punctured version is denoted as rm of r comma n star. Okay. So, for instance, the 743 Hamming code will be rm of 1 comma 3 star. Sorry. Yes. Yeah. The first bit will be drop. Yes. How can it be a cyclic code? So, what do you mean by which system is cyclic? All one system is obtained by a cyclic shift of itself. Right. So, see, the way I think of cyclic codes is you have a code word. You have a code word. If cyclic is shifted, it should continue to be a code word. That's all. Don't think of obtaining the code word as a cyclic shift of something else. And that does not really need it. Okay. Now, the cyclic shift of gfx. See, multiple of gfx. So, you have to add also. Okay. Is it okay? You see, I mean, it stands as a difficult result to prove in general. So, I am not going to prove it. So, what will be the parameters of rm of r comma n star? See, the rm of r comma n is a 2 rm comma k of rm comma 2 rm of r2. Okay. So, rm of r comma n star is equivalent to 2 rm minus 1. What will be the k? k will remain the same. Okay. So, you can quickly see that this will be true. Okay. So, k will be the same. Okay. So, you will have the same number of polynomials. It's not really difficult to see that. Okay.  Okay. So, you go around the way. You will have an out-weight vector. You can show that's all. Okay. And this can be a cyclic code. Remember that. That's an advantage. This can be made a cyclic code, which means it has a generator power on it, GFX, which you can use in a systematic encoding. Very nice. Okay. Okay. So, what is nice is, again, you won't see this result. It turns out, for GFX, both code, there is a formula. What do I mean by formula? Some way to generate GFX. Okay. Somebody asked me, what is the formula? If you want, you can see our book, the Linen Kostile book, has this formula. So, you can see it. It serves. It's a complicated formula. You're not going to do the derivation class. But there is a formula. Okay. So, it's not very scary. Somebody will tell you how to find the GFX for everything. Okay. But it's not a BCH code. In many cases, it's not a BCH code. In some cases, it is a BCH code. Very few cases. Mostly, it's not a BCH code. Okay. It will be a subcode of the BCH code. Okay. So, it will be a cyclic non-BCH code. Okay. And let me give you an inclination as to why this has to be true. Okay. Why will this thing work out to be cyclic? Okay. Remember what I want to show is, if I have a code word, every code word corresponds to a polynomial evaluation. Okay. Now, this polynomial evaluation is basically evaluation at alpha power i. If I cyclically shift it to the left, what happens? I'm evaluating this side. Alpha power i plus 1, in a way. Right? Right? I have to show if I do f of b1 through vm evaluated at alpha power i plus 1, it is the same as some other G of b1 through vm evaluated at alpha power i. Right? If I show that, I'm done. But that's how you do cyclic. Okay. So, it turns out this function, instead of thinking of it as a function of binary variables, you can also, if it's polynomial, instead of thinking of it as a polynomial with binary variables, you can also think of it as a polynomial with coefficients from G of 2 power i. Okay. So, it turns out it's possible. It's a very complicated, higher quickly polynomial. It's a bit more complicated to explain it to you. So, you can think of it always as a polynomial over G of 2 power i with coefficients from G of 2 power i. You put the argument alpha power i directly into the polynomial. You'll get back a bit. Okay. You'll have a polynomial over G of 2 power i with G of 2 power i with coefficients. But when you plug in every alpha power i, what will you get? You'll get a bit. Okay. You'll never get another element of G of 2 power i. It's a crazy kind of polynomial. Right? Think about it. So, there are polynomials like that. So, it's quite advanced to think about what these polynomials will be. But there are polynomials like that. So, you can think of this polynomial evaluation not just as a polynomial in multiple variables, but binary values. You can think of it as a polynomial with a single variable that that single variable comes from G of 2 power i. Okay. Once you go to that kind of thing, multiplying by evaluating an alpha power i or i plus 1 is very simple. You simply take, you take S of x and put, instead of x, you put alpha x. Okay. You'll get a G of x, which is S of alpha x. S of x evaluated at alpha power i plus 1 will be the same as G of x evaluated at alpha power. Something like that. Okay. So, it's very easy to go once you have a polynomial over G of 2 power. Okay. So, we won't see the proof, but that's the idea. So, what I allow you to do is, in the BCH and RS codes assignment, there is a last problem. You go to my web page and look at BCH codes and RS codes. There is a last problem. That problem basically describes the construction of each element codes by polynomial evaluation. Okay. Try that for a while. Okay. Instead of going to the read mail, of course, it's much more fancy. Multiple variables and all this. The each element code is much easier. You can try solving that problem. Okay. There are things for that problem. Try solving that problem. We'll see exactly what I mean. Okay. There is a way to show each element codes, acyclic and all that. So, it's nicely come out. Okay. Try that. And that will give you, I think, a test of how this proof will work. Okay. So, in general, if you're interested later, come and talk to me. I can give you references. You have to see it. Okay. So, that's the idea. So, essentially, the main result we have is quite useful and powerful. Okay. The punctured version is acyclic. And it has to generate a polynomial which you can compute. Somebody knows the formula for it. You can compute it. So, it's six variables. Okay. So, you can do encoding very easily. Okay. All right. So, that's the read mail encoding part. We'll, as far as encoding is concerned, we'll simply start that. We're going to move towards decoding. And, okay. So, of course, the main reason why I did encoding is to convince you that the encoders are simple. Okay. We're going to have a very large read mail encoder. Encoding is very, very simple. And the next thing we're going to talk about is decoding. Okay. So, the decoders are read mail encoders is really, really interesting. Okay. So, it's one of the most interesting and useful decoders. Because it will be a decoder which we have never seen so far. So, a very implementable decoder which can correct errors beyond minimum distance. Okay. So, if you think about it, it's a scary kind of. Okay. So, we never saw a decoder like that. So, it's a very simple decoder. But it's probably to you. We'll say it's the simplest thing in the world. It's much more simple than all the belly cam, massy, you know, some era locator, whatever, the erasure, and all this. No such conclusions. Very simple decoder. But it turns out it can correct errors beyond minimum distance also. So, it's powerful. But on the flip side, it won't have, it won't correct all the errors that it should be corrected sometimes. So, if you have an error correcting capability t, less than t, it might not correct. Okay. So, in many ways, it's a very interesting kind of decoder. Okay. And we'll see in the next course, we do the next course. Many of the decoders that we'll see in the next course will do like that. Okay. So, they will not, they will not have any guaranteed error correcting capability. But they will perform very, very well because they can correct much larger number of errors most of the time. Okay. So, that is the idea. So, it's an interesting career. And we will the first kind of decoder that we're seeing that that plays a role. Okay. So, I'm going to first describe the concept behind the decoding and then tell you how to implement that concept for your molecules. Okay. So, what is the concept behind decoding? This concept is called majority logic. So, you decode the majority logic. So, let's say we have a code C, which is an NK code. Okay. And I have a code word C1, C2, Cn. It is sent over some channel. Right. And now if let's have it correct, whatever it is, we will see R. R will be R1, QR. Okay. So, the first difference between the decoder and boundary distance decoders and all, we will not be attempting to find the entire code word directly or entire error vector directly. We will try to find the decode bit by bit. Okay. That makes, as it turns out, we're suppressing that it makes a big difference. Okay. So, whether you try to find bit by bit or whether you try to find the code word directly. Okay. So, in a way, if you look at the boundary distance decoder or even the syndrome decoder. Right. We're trying to find the code word directly. Right. We're trying to find the error vector directly. So, what we will do is, we will be trying to find every bit. Okay. I will be only concerned about decoding C1. Okay. In fact, what will happen is, we won't even try to make sure that all the individual bits that we decoded together will actually be a code word. We don't even have to care about that. Okay. So, we will simply say, I decoded these bits. As you know, it's systematic encoding. I will simply take a subset of those bits and say, that's my message. Okay. I don't even have to worry about ensuring that the entire decoded code word I put out was actually a code word. Okay. So, that is called bitwise decoding. Okay. So, this is a bitwise decoding method. So, we will try to find CI cap for every item. All right. And like I said, there's no effort to ensure that C1 cap, C2 cap, so on, till C1 cap is actually a code word. We won't even try to do that. But we'll try to find each bit of the code word. Okay. Okay. Estimates for each bit of the code word. Okay. So, that's the first idea. Okay. So, the next idea is, again, these are all powerful ideas, which you'll see later on in the next second course, that people have used to build very powerful decoders. Majority of it is not that powerful, that the concepts here are usually useful and more in general. Okay. So, first and bitwise. Okay. So, what is really nice about bitwise is you're not worried about N. N could be very large, 100,000. But I'm only worried about one bit at a time. Okay. So, maybe it's not too scary. Okay. The next idea which is usually useful is, you use some local structure around the ID code. Okay. So, what do I mean by local? I'll come to it. Okay. So, the local structure can be defined in so many ways. Okay. The local structure is used to estimate CIF. So, these two ideas have proved to be very powerful today. Okay. So, in the modern interpretations of codes, doing bitwise decoding. And how do you do bitwise decoding? You use only the structure around CI, so to speak. Local structure. Okay. So, again, when we go to a very large block length, local structure is much easier to deal with than the global huge structure. So, it takes like 2 power N minus K and all. You won't even see. Okay. Can you agree to go one bit and what's around that bit? What do you mean by what's around that bit? Let me clarify that. Okay. So, my idea is, at least in a very logic decoding, this is how we interpret this local structure. You look for parity checks that involve CIF. Okay. So, this is the important thing. And this is used even in modern codes. You always look for parity checks that involve the ith bit. Okay. So, that is local structure. Okay. What do you mean by parity check? What is a parity check? Every way of the parity check matrix is a parity check. Are those the only parity checks? No. Every codeword in the dual is a parity check. Okay. So, when I say parity check, I mean codewords in the dual. Okay. So, whatever, when I say codewords in the dual, that parity checks that involve CIF, what should this codewords in the dual have? Right position should be 1. Okay. Okay. So, we have this read Miller code, we are stuck to decoding it. And how do we do it? We try to exploit local structure of read Miller code. Okay. So, another interpretation what people do is, people don't care about starting with the read Miller code. They will say, I will define a code with very nice local structure. Okay. I will, as it is, when I define the code itself, I will define a local structure that is very nice. Then you don't have to go hunting for codewords in the dual with one and right position because you demand it like that. So, you will just see it and you will pick it up. Okay. The read Miller code, since we are doing, like, some kind of a legacy compatible implementation in a way, so you have the read Miller code, you are stuck with it. You don't have a choice of defining your own code. So, you take the read Miller code and then you try and find its local structure. What is the local structure that you can find? Try to find codewords in the dual, which have a one and the right position. Okay. So, now I want to remind you of one powerful thing about the cyclic property that you saw. Okay. Because you have the cyclic property, what is nice is, at least for the punctured version, if I find local structure for the right bit, let's say that I equals one. If I find local structure for the first bit, what have I also found? I found local structure for every single bit. Why is that? You simply cyclically rotate it. Well, you know, dual of a cyclic code is also cyclical. Okay. So, if I have a palliative, if I have some codewords in the dual, I simply cyclically shift it. I'll get at the codewords of the dual. And because I cyclically shifted it, all the ones in the first position will become ones in the second position, third position, whatever. Okay. So, that's where read Miller code is nice. So, you have to find local structure for a particular bit. The same structure will be applied according to the cyclic shape. So, if you cyclically tackle with a cyclic shape, the cyclic order comes in with alpha, alpha, r i. Okay. And you'll see the local structure might be easier to find than the normal bit order. Because 0, 0, 0, 0, 0, 0, 0, 1 next step. Okay. So, when you do the cyclic shape, there is slightly tackle. As long as you tackle with that, it will do. Okay. That's the idea. So, these are the partial ideas. Like I said, they are very, very modern interpretation where the tool to be very useful. But we'll apply it to the read Miller codes first. So, we'll try to find parity checks that involves the i. And I'll tell you what majority logic is. Okay. So, that we'll do in the next class. Okay. Sometime next week, in the last class for this week. Enjoy your Prashatabh break. I don't know if you're doing something interesting for Prashatabh. Anyway. So, have fun. We'll meet on Monday.


Russian attack on NATO looms as Trump threatens to abandon alliance. With Donald Trump threatening to pull back US support, NATO countries grow anxious over forecasts that Russia could rebuild and invade alliance territory within coming years the New York Times reported. We have always kind of suspected that this is the only existential threat that we have. Major General Vaiko Velo Parm, Estonia's land forces commander told the publication. According to Parm, many NATO militaries are not ready to fight Russia. NATO has worked to revitalize Europe's military capacity since 2014 when Russia first invaded Ukraine. But the progress made could still leave allies unprepared, so it's not very comforting. General Parm said, as NATO stages its biggest exercises since the Cold War, anxiety simmers that the drills ready forces for imminent attacks. We have to realize it's not a given we are in peace. Said NATO military committee chairman Rob Bauer, analysts say Russia could rearm within years of the Russo-Ukrainian War conclusion. Just because it got all chewed up in Ukraine doesn't mean they're off the board for a decade or more, said Atlantic Council's Christopher Scaluba. Norway's top commander echoed the distress of being short on time before unpredictable Russia's attack. There is a window now that will perhaps last for one, two, maybe three years, where we will have to invest even more in a secure defence. General Erich Christopherson warned, this rapid timeline combined with wavering US support if Trump returns to power leaves European allies bracing for earlier aggression.


Hi I'm Maureen Feynman. I'm Heather McHugh Brun. And the topic of this week's unit is garnet. So garnet is a silicate mineral that grows at high pressures most commonly in the interior of the earth. Garnet is most commonly found in metamorphic rocks which are a little less common at the surface of the earth than some of the other types of rocks and minerals that we've been looking at so far. Silic the garnet is a very hard mineral. It has a hardness depending on the specific chemical composition of between six and a half and eight on a Mohs scale. And that has a couple of implications for the use of garnet. One is that garnet is relatively difficult to break or cut although it can still be broken into chips or carved carefully into shapes. Another result of the hardness of garnet is that it's really useful not only as a material in art but also as a material for producing art. Hard garnet will form an abrasive that can be used for sanding grinding polishing and it can also be used to make tools for cutting other less hard minerals or rocks. In art what we're going to be seeing in this particular set of lessons is that garnet is used primarily in jewelry production and it really becomes available sort of in the late classical period when Alexander the Great and his armies pushed into the Indian subcontinent extending the territory of Greece and well well into the the eastern world and this opened up trade routes for garnets and suddenly shortly after Alexander's campaigns we start to see the emergence of all of these incredible garnets in jewelry and that's something that's going to be addressed by one of our short video clips with Adam Levine from the Toledo Museum of Art. He presents a number of objects to us and we'll sort of trace the the ways in which these garnets were used in art and the symbolism people tended to really favor the red garnets and even though these all look sort of black to you here we're going to insert an image so that you can see that under light and when cut into thin sections these actually shine sort of a deep blood red and this could be used in the medieval period for example as a symbol for blood.


Yes, we're at Stets, the home of champions in the bread basket parish of St Elizabeth. It's another serving of school board football in the island of Jamaica in the DeCosta Cup for 2023. And this is on your home of champions. As that shot shows you, we have a bit of rain, but we'll be we're gonna be seeing a magnitude versus Roger Clark. And I was about to ask Christopher Taylor, the analyst, what his prospect for the weather was, but here's a gift as a meteorologist will not be tested today. That's the magnitude team you see it on screen. They're getting their feet wet proverb as would say proverbially. And we're getting ready to see a team that hasn't really had a great tradition in schoolboy football, Roger Clark. And let's see how they fear in this encounter, they've had measured success, especially in the 2018 2019 seasons. We'll hear more about that in a short while. But we're getting ready to see the teams march out. Officials getting ready. So they enter the field, which has become synonymous with schoolboy football in the island of Jamaica. Centres by technical two finals were completed at the senior level last year on this very ground. And for 2023 or first the trip in the bread basket parish. Roger Clark led by their captain, John Nickens, meeting the players from Magity.  So the referee, Mckina Brown, assisted by Princess Brown and oral right. Captain there for Magity. Daniel Martin. And they have their confabulation now. The referees and the fourth official. Here's a team sheet now for Magity High in goal. Jero James, Giovanni Findley, Dante Blair, Akeem Wrangley, Chadric Jackson, Shawnee Daly, Rashane Lewis, Oswain Miller, Jarene Robinson, Daniel Martin, and Rabina Bart, their coach by Xavier Hendrix. They'll be playing with a four three three formation. Magity High, Daniel Martin, their leading goal scorer with two goals. Rabina Barrett off the mark with one as well. And look out for the creativity of Rashane Lewis, who has two assists. Here's the team sheet for Roger Clark, their goalkeepers Rickon Powell, to Kwan Hansen, Rashane Malcolm, Dante Blake, Deshaun Wright, Jamali Wilson, Adiana Poiser, Kimani McLean, Anthony McDonald, Rodain McDaniel, and Jordan Akeem's coach by Errol White. Four, two, three, one, the formation for Roger Clark. And their only goal scorer, Kimani McLean, who has two goals, their number 10, look out for him. And he will get a lot of help from Othniel McDonald, who has the only assist for them. Their number 11, those two goals coming against LaCovia. Magity trying to play that one deep up field. But that's a goal kick in the early exchanges. Magity trying to settle in the half of Roger Clark. Daniel Martin has it. Still Martin playing it back now to his defenders, Findlay and Blair, all the way back to the keeper, Jeroi James, Blair, to Jackson. Seeming to have some problems with their foot in the players. Slight drizzle, but only compounding the measure of rain that St. Elizabeth has been receiving, especially Santa Cruz, where they're continuing. Number 10 plays it, that's Rashid Lewis, onto the path of Jureen Robinson. Fires her, but that was blocked fortuitously by Roger Clark. Looked to have been taken on Hanson there with the crucial block as a Magity on the ascendancy in the first three minutes of this encounter. They have a corner kick delivered, a bit short, played out of touch by Radane McDaniel. So look at the coach there, Xavier Hendrickson. Hendrick's rather for Magity, math teacher. So the match commissioner has made the decision to halt the proceedings based on lightning. So that's a break in our coverage here at St. Elizabeth Technical in this encounter between a Magity and Roger Clark. As I said earlier, came being paused because of lightning and the teams are being instructed to go back to the locker rooms and the safety must come first. So we're back inside this encounter between Magity and Roger Clark. Game has resumed, that ball is Magity. So still in the early stages of this game, of course the game was paused because of lightning strikes and after a lengthy wait the sun is out in all its glory. So he's a free kick, deep, able to play it out. Still trying to get the ball out is a Roger Clark. Magity now on the ball. Rache and Lewis seems to be a bit injured but Magity still has the ball. So the referee halted proceedings there. Rache and Lewis, the striker for Magity down. So still Magity. Referee there giving a kick. Free kick to Magity. Just outside the 18-yard box. Here's a kick way over the top and that will be a goal kicker for Roger Clark. High this in the seventh minute. I haven't pressed anything. So Roger Clark with a throw in that was wasted. Magity now able to clear. The ball played through. With the Jureen Robinson now. Empties the box. Jereen Robinson. Oh he buffed that one. It was under some amount of pressure but he really should have done much better. The number 12 for Magity. Roger Clark they collectively breathe a sigh of relief that in the ninth minute of the encounter. Has some speed about him Jureen Robinson. Just wondered if he should have probably cut in a little bit more just to close the angle. Just to open up the goal a little bit more for him. An important nudge though in the defense line of Roger Clark. Preventing him from getting the accuracy needed on that shot. Here's Magity. It's to be Lewis. Still Magity trying to advance. Chadwick Johnson now turns inside his marker. Lays it off and gets the return pass. Back now to Blair. He goes all the way back to Findlay. First real percussion of passing from either team. The surface even amidst the rain doesn't seem as soggy as we would have seen in past years Christopher Taylor. Yeah even though there are serious thunderstorms in terms of lightning. I wouldn't say that the rain was heavy for any prolonged period. Yes a bit consistent but on the lighter side of things so I think that's one of the reasons why as well that we haven't seen the heavy remains. Robinson the intended target he gets it one time. Oh dear me how did he manage to miss kick that one. Still Robinson. Cues it for Lewis. Lewis just wide of the target and you can see that Paul in the goal for Roger Clark would see nothing about that one had it been on target. But yes football is on show here at Santa Cruz. Another bite strike by Lewis. Has scored so far this season to assist to his name though Rochaine Lewis and that shot skewing off the outside of the boot. There you saw the wet condition so the ball going a little bit away from him didn't wrap his foot around it but look at this. Good first touch getting inside of the phenomenon needed to use his left foot there. Here they come again. Oh that one went over the top. Has Wayne Miller there the culprit on this occasion and not able to bring it on target. And you just get the feeling Christopher Taylor that soon and very soon if Magadi continues this vein of pressure this vein of form they'll be able to open the scoring. Yeah lots of defending for Roger Clark they have conceded 18 goals in their three games as far so clearly under pressure generally in their games and quite frankly they haven't been able to stop the teams from scoring. Here's Magadi once more. Lewis to Robinson that's generally been the pattern of progression. Lewis their essential attack in midfield position spread it wide to the winger. So if Hendricks they're giving some instructions it's not sure if they'll change that pattern of play. It has reaped some amount of dividends so far but the finishing just hasn't been as polished as I'm sure he would like. Roger Clark they're trying to clear but Magadi cannot throw in there on the right hand flank. Jackson takes it quickly to Lewis stripped of possession but Magadi regains it and foul there against Young number nine from Roger Clark, Adriana Poiser. Interestingly we spoke to coach Errol White and he mentioned that there are seven players under 16 for Roger Clark and four under 14 so he's definitely trying to build a program at Roger Clark High. For those who might be wondering, formerly Balaclava High School but it was postlessly named after Roger Clark after his death in 2014 this school that the name was changed in 2015 here's daily. Magadi still with the ball Oswayne Miller there brought down Tecwan Hansen there the number two for Roger Clark the guilty culprit. They have a free kick can be from a very dangerous position so they line up with a three-man wall it seems Roger Clark Rabina Barrett the man standing behind this one and this is perfect territory to go directly to goal but they also have a lot of coverage at the back post does a Magadi High School. Barrett steps up way over top. Yeah this is a point in strike in the end and he knows it as well you can see it on his face with Magadi in his first 15 minutes of play.  Poor clearance if you could even call it that here's Magadi once more Findley brought down the number nine Ashani Daley. Yellow card coming out as well and McLean there the only goal scorer for Roger Clark since the beginning of this season here's the opposite number 10 lifts it up and almost very dangerous there just able to claw it out with Roger Clark it's a throw in for Magadi taken quickly by Jackson trying to find Daniel Martin but booted out of touch. Magadi only better off by a point of course earning a draw against Monroe College in their opening game of the season two one losses at the hands of St Elizabeth Technical and B.B. Koch their other results tight matches but coming out on the losing end of things which were positive so far in their approach. If they come again Magadi stripped of possession even but Donald loses possession for Roger Clark daily sends it across to Martin. Martin it was always going to happen that's Rabina Barrett from Magadi High School opening the scoring in the 17 minute. The writing was always on the wall and they have opened the scoring against Roger Clark. Beautiful pass all the way through to Barrett did well and he finished with a plumb rock one. Powell had no answer to that one at the near post. Good first touch good finish by Rabina Barrett it's his second of the season and yeah I assure any gymnastics instructor will give him a high grade for those backflips as well. Barrett but as you said Dean certainly the writing was on the wall for Magadi attacking well putting the pressure on Roger Clark and that cutting-edge pass really opened them up. Slight shift in there for me well not necessarily the formation but the assignments of different individuals we see the number 12 who was on the right you can shift it to the left but Roger Clark they're trying to conform that's broken up by Anglain so the assist came from Ashane Daly for that goal converted by Rabina Barrett so Blair to Findley to Daly the captain there getting a touch that's Daniel Martin in number 13 they're trying to impose their possession based game on this encounter hasn't been very hard to do so the youthfulness being experience of the Roger Clark team throwing for Roger Clark Daniel takes it to Poiser Adriana Poiser the young number nine turning release now for the goal scorer Rabina Barrett Magadi advancing that touch was a bit too heavy but fortuitously it still fell and it's a goal to his feet and he buried it that in the 20th minute 17th minute 20th minute the floodgates have been open things falling apart here for Roger Clark this touch has seemed very heavy it was heavy had a lot of speed about him though the goal scorer Barrett and then coming off the upright and Miller the first to react and miss at the back for Roger Clark just here where the ball hits off the post just taking too long to react skipping out of the way and yeah Miller certainly made the most of it his opportunity his first goal of the season distance now between Magadi and Roger Clark here's Magadi that's daily trying to bring in Barrett but that ended up in the hands of the goalkeeper Raquon Powell unable to really clear effectively almost a good turn that but it's still with the Magadi now he advance here's Robinson Robinson sends it across oh my he just couldn't apply the finish there and they're turning on all the styles this Magadi team and they're all smiles as well beautiful pass that from Martin to Robinson whipped it across and that's when Miller just couldn't get the final touch there to bring it on target here's Magadi daily that's broken up but they regain possession Miller now Powell should handle that one easily does Powell tries to clear been doing an excellent job in clearing that only compounds the pressure on Roger Clark ball being played out now now it's with Roger Clark dispossessed back now to Wranglin make that Wranglin the number six Martin the captain to Lewis Jackson see the difference in size there just easily able to push off Poiser here's Barrett now on the right and flank unable to get by his marker Jaden with Jordan Aikins for Roger Clark another change being made Robinson now returning to the right and flank early in the game he really had some incisive runs on that flank went to the left didn't see so much beauty coach Hendricks has brought him back to that near of the park let's see if it rips dividends for him personally hasn't really been involved in any of the goals per se here's Barrett unable to well he gets it back Barrett Lewis is always suspect from that angle his positioning unable to really bring a touch of finesse on that one well I tell you Cristela I'd have never guessed that the Sun would be out in this measure of splendor with all the lightning we had earlier I wish you would have you know taken up your meteorological background and give me some predictions concerning that I'm so glad you didn't need to come to me for any confirmations but yeah great to see the Sun out now hard to believe that the skies are looking so good considering what it was like and all ago everyone shielding for cover from the serious fork lightning strikes I tell you what as a parrot sent Elizabeth rejoices because big match coming up a bit later on big sent Elizabeth Darby Monroe and Stets will renew their rivalry and yeah the crowd it's just a foretaste of that I'm sure it will grow as the afternoon progresses you can actually see in the distance there is the Glen the Glenmere football team as well the Costa Cup team here watching in their full red have a day off today the Glenmere team and they said well you know why not come and watch something to Elizabeth football of course there are two parishes away in Clarendon our Glenmere two parishes to the east you would say I wouldn't really even bring in the parishes now because with that new highway they're only 20 well not 20 but really just about 50 minutes away now yeah still two borders away though well you are right but in terms of time ease of travel yeah I see you certainly enjoyed the travel from Kingston today to St Elizabeth who wouldn't on the new road who wouldn't especially when it's free not that costs would have been primitive to me I'm not paying by the way listeners here's magnitude they're playing deep in their half and Roger Clark they're not afraid to really push up that's a level of confidence that being two goes down so early I wouldn't have expected but here comes magnitude it's now with Miller he missed one just a few minutes ago but he had the composure to finish this one with the plum in the 20th minute Oswain Miller brings a score and Errol White must have a lot on his mind yeah nice built up again in the wide areas into Miller who scores his second of the afternoon two goals eight minutes apart probably should have done better goalkeeper Powell to his right hand side his near post and just sneaking by him but Miller came into this game hadn't scored so scored so far in the season and now he has a brace wow but it is said that you should be okay while the sun shines and the sun is shining on magnitude right now yeah things have been very comfortable for them controlling the tempo of the game the position the midfield player and certainly they look dangerous in the wide areas well coming to poiser unable to take it first time still has it poiser forced wide by Jackson got by only just I love to see a diminutive player with the tricks and frills but here comes Robinson now goalkeeper came forward had to retreat Robinson just off target it would have been a spectacular goal a spectacular attempt in the end but bit of curl just brought it away from the far upright good attempt by Robinson but as you can see that touch as well taking it away from goal not a bad strike in the end but the natural curl of it taking it past the upright but still would love to see Robinson cut in towards the goal just to open up his opportunity in terms of shooting he's been he was switched to the left-hand side of the attack and it looked like that might have benefited him more because he could cut it on to his right foot he's back on the right now and yeah another opportunity he's had about three overlapping runs I make it so far he's touched just before the strike has always let him down coach Hendricks given the baton can shoot it's down for Roger Clark that's the number nine Adriana Poiser it's the Glenmure team there's a lot of expectation on Glenmure road that's the number 10 Kyle Gordon what a play was in the Ben Francis cup final was the man of the match there three victories from three matches so far for Glenmure coming off was it a 15-0 win I think it was 16 16-0 win against Poros you have a big match coming up on Tuesday against Denby right that should be a good one Dunstan Cohen I was just about to say yeah coming off his what nine goal exploit yeah that must be a record Errol White there before the match had a chat with him actually started coaching out of pure necessity really that was during his time at the business house level with the Jamaica Railway Corporation coach was unable to continue and everyone said why you're the next coach and he was fortunate to have been in the same place training at the same field as Jackie Walters separate team and he learned a lot would watch them in the morning then replicate that in the evening and for his credit would have guided LaCovia to semifinals here's Magati Jackson that's in Jackie Walters we'll probably see later yeah coach of Monroe yeah so as I was saying yeah he would have brought LaCovia in his time as the head coach of LaCovia from 1995 to 2016 to the Dacosta Cup semifinal in 2008 where they lost to eventual champions in James High and also would have gone to the Ben Francis semifinal lost the Spalding and in 2015 would have advanced that team to the Flow Super Cup so has had some measure of success for LaCovia Errol White and he's trying to bring that same quality engineer that quality at Roger Clark and I use that deliberately he's an auto mechanic teacher by training and profession yeah interesting German automotive and with a young team like this obviously they experienced again at the Dacosta Cup level invaluable there's a corner kick so hopefully if these youngsters stay together in a year or two they could be a force to reckon with yeah with seven 14 players under 60 maybe that 11 players under 60 yeah could really do some damage if he can keep them together and if they remain focused development isn't always linear though but it's certainly not an unreasonable expectation here's Robinson yeah heading the ground again Robinson tells me in his team the corner interesting choice that by the captain John and Akins here's Jackson who whips it across daily they're trying out the volley would have been spectacular had it gone on Shawnee daily the number nine daily back to Martin to Miller Miller gets the return ball or does it know it looks to be Robinson he switch flanks again daily now daily from distance and a deflection there reducing the sting of the attempt it's a corner kick and the number 15 goes over Rubino Baratheon open the scoring in the 17 minute here's a corner kick coming in flicked on here's Magatine Roger Clark trying to clear and they able to do so should be a throw in for Magatine Bradley daily Lewis spreads it across Robinson turns Robinson back to Lewis Magatine Lewis from distance easily well I almost said easily handled by Powell I guess that was predictive and not the reality of what really happened hasn't really had the the best glove work Recon Powell the goalkeeper Miller to Barrett Barrett comes in Barrett still to Lewis now cleared easily it's really been one way traffic friendly trying to bring that one to Robinson but that was cut out Poiser who was down earlier is now fighting Martin daily he has a smile could have been dangerous for Roger Clark Robinson to get it a crucial touch to bring it across goal Powell though able to to avoid the danger peanut Barrett was charging in player boots that out Lewis Robinson he was offside poor clearance again emblematic of what we've seen of Roger Clark in the defensive third let's see if they can have some joy now they come forward McLean loses possession rather cheaply Miller Martin Wrangling back to Miller to Barrett that was deflected out for a corner kick then in the 49th minute yeah not much time to go just about five minutes to the half didn't see the fourth official though put up two minutes not sure why Martin so it seems the half being played for 40 minutes punched out by Powell bit of redemption for him didn't look comfortable but got it done I guess that's the important thing for him Powell yeah I guess because of the the interruption due to the weather and the fact that there's a second game the officials have decided to reduce it from 45 to 40 here's a corner kick into the side netting Roger Clark card they are shown to the Charlie daily for that challenge on Kimani McLean the number 10 for Roger Clark who was really advancing in a dangerous position instructions been barred by the keeper J. Roy James from Agati that one just deflected off the wall so both halves 40 minutes for total of 80 minutes for regulation time given the circumstances and the the fact that there is a another match I can understand and the referee indicates the end of the first half in Santa Cruz Santa Elizabeth Magati leads Roger Clark three goals to Neil and it's been one-way traffic at the Santa Elizabeth complex Roger Clark they really haven't been able to develop in a great pattern of play but the experience should do them well Magati they're trying to rebound from some not so hot performances early in their competition this season and this performance must be a relief for Xavier Hendrix their coach yeah you hear the music blaring in the background it's it's a schoolboy football the Dacosta Cup Magati three Roger Clark Neil we're at the half the water break fans from the stands I'm here with Linvo Linvo just give me a quick analysis on this game Magati versus Roger Clark it starts a bad game you know I can see you can see the superior team which is Magati and Roger Clark they're trying you know but I think Magati have the best at play passing the ball around quickly and enjoy we're having fun here it's a nice of a game despite you know the the Magati team is the most superior team but we're enjoying again all right cool so let me know um if you were the Roger Clark coach what would you say to them in this half-time break let's come out and play have fun you know let's try to get engaged in the game and see what can happen anything can happen is a it's a game of uncertainty so let's go what I'm playing have fun give me a score like prediction there's three nil now but will they be able to get more goals Magati well we look we're probably looking at six zero but probably Magati probably um the Roger Clark team probably get lucky and I'm pulling one so you never know that's football all right cool let's head back now into the game for the second half thank you so much Gerard ready for second half action Magati versus Roger Clark three goals to nil the scoreline at the start of the second half let's see how Roger Clark can respond or how Magati can compound what they've started referee Makina Brown a bit unsettled some interference that's the referee Makina Brown ready now for the start so play resumes now Roger Clark trying to push forward there the man Rashane Malcolm is brought down this distance certainly within the reach of Kimani McLean they're number 10 but we're still waiting the first shot from Roger Clark whether on target or off so this is this is a good sign for the second half for them to remember we are in the second half because it's 40 minutes a half so that it's an 80 minute game due to the interruptions McLean well did get the first shot it was off way off he actually has a goal from a free kick let's see if he can get his shooting ball boots on sooner rather than later he's the only good score so far for Roger Clark Clark scored both of them against Lakovia in that 10-2 loss suffering for Roger Clark just at the halfway line poiser still being able to get by players diminutive number nine brought it out of touch Magati now able to reset and build on their passing game their passing was fluid in the first half you have to give them credit for that it did seem very organized having a clear direction as to what they were trying to accomplish and let's see if they can get some more goals to complement the efficacy of their passing so through into Roger Clark taking quickly taking only for the benefit of Magati yes Magati wants more Blair effective turn that from Jackson the number eight Delea space I wonder if he's moved from that position Cristela whatever we see him on camera he's right there he's a cool head calm on the pressure there the defender from Roger Clark Dante Blake Magati's still trying to attack Robinson now it's seeming with Barrett Miller that was far offside from Barrett though certainly wasn't paying attention to the line the number 15 and clearly offside this ball watching a bit and in guilty on that occasion Robina Barrett who actually picked up the assist for the third goal as well to go along with his 17th minute strike in fact he's been involved in all three goals because the second goal as well it was he's over dribble if you want to call it that ended up on the upright and came back to Miller who executed the finish here's Magati he's offside again jinx perhaps he's watching the sports mix out yeah Xavier Hendricks came to the school in 2018 and that's where they have seen their greatest success Magati quarterfinals in 2018 the quarterfinals again in 2019 and actually in that 2018 went to the semifinals of the Ben Francis before losing to eventual champions Charlie Mont that season way back in 1998 as well they did make it to the quarterfinals Magati but that two year period in 2018-2019 quite strong to have a young team Magati in a rebuilding phase lifted across Robinson at the end should have really done better had a lot of time Robinson and with all the pace he has he really just hasn't been able to deliver a final blow a telling blow and in that 28 season 2018 season where they figured maybe they could go even deeper they ended up in a tough quarterfinal group the likes of Manchester who they earned their dragons but then Cornwall College and Glenmere was it no Garvey Marseille yeah tough matches they ended up falling short Martin would fancy a shot from this distance Martin is a second look at it the angle seemed a bit more glorious than the eventual shot the possibility Roger Clark has a metric modern-day statistics really just from the eye test can't see any progressive defensive pass they've all gone to the favor of Magati Robinson they are offside as well so most of the passes have been very low unable to clear the lines really get it up in Magati territory here they come now McDonald unable to lift it over his marker calm composed in playing out from the back in this particular encounter is Magati Martin Lewis yeah willing to share the ball Magati that has been a highlight of their play everybody getting a touch no one trying to overdo it calmness in the back and patient build-ups when they do decide to attack though fluent on the in the flank areas and a lot so speed that's where they've really opened up Roger Clark look at their their coach Xavier Hendricks his favorite coach Pep Guardiola so no surprise that he's interested in that kind of play the Magati coach and actually in his time of playing he was a central midfielder believe it or not Xavier Hendricks played his schoolboy career Monroe so I guess he will be hanging around later to see what his alma mater can do against the probably the top team in Saint Elizabeth at the moment that game should really be a cracker of course in the technical they won the all island under 16 competition last year and they were finalists in the rural area under 14 but they won the under 14 as well they lost the all island title yeah to Jamaica College and I tell you that Jamaica College under 14 team it was a joy to watch them Cristela but yeah and defeated champion in the under 16 competition yeah they've been a bit too hasty in their advance in the final third Magati just by the eye just about three offsides in the last five minutes yeah anecdotal and quote me on that one but yeah it's a total of five now but three of them certainly has been in this half alone getting ready to make a substitution getting ready to enter the game is number four Kimran Mullings and if you hear the horns the home team they're getting ready to warm up and the gallery shouts their own phrase here they come now Robinson Powell really should have done better earlier it's left loose Robinson yeah he had to score finally he's on the score sheet Rick one Powell really did a haste job there really should have come out with a greater level of conviction but yeah good offensive thrust from the number 12 Robinson and he was on hand and where Powell failed he delivered yeah well Robinson has had quite a few good tries and one thing he's certainly been good at is his explosive speed in the wide areas this time he's tail alert Robinson and as you said Powell a weak attempt the goalkeeper yeah not strong enough and not getting his body behind it as a second line of defense the rebound falling fortuitously to Robinson his first of the season though Jareen Robinson and now four for Magati well their goal difference looking much better should it go to goal differences they'll definitely want to pile on some more because Lakovia scored 10 against esteem here they come again all the way across Robinson on the end of it couldn't bring it back into play it goes out for a corner kick and Oswin Miller also that ball sprayed across by beyond him that would have been a hat-trick if he was able to get it on target Miller so it's a corner kick for Magati taking shot dispossessed and Roger Clark trying to advance to throw in Kwan Hansen looks a bit labored here they come again Magati Miller Robinson took it down well burst of speed into the box Jareen Robinson Powell got a hand to it here they come again Martin to daily afraid to use the width of the field is Magati seem to be a free kick for them just outside the 18-yard box for the right of the half space well your friend Mr Miller is still leaning against that post head coach of Roger Clark just looking on so Robinson is off and coming on Kevoi Barnes so having scored that goal Hendrik says coach Hendrik says let's see someone has a chance let's see what Kevoi Barnes can bring to the game Mullins also coming on for Wrangling here's a free kick lifted up beyond everybody it's a goal kick for Roger Clark here they come again Magati going straight to the goalkeeper Powell again not very convincing in that take to his credit only four goals conceded so far in this encounter Miller and he was outside young Lewis there the number 10 yeah that was definitely a close call still didn't manage still didn't manage to finish though Lewis and at the time he wouldn't have known he was offside disappointing in terms of that in the positive now in terms of goal difference Magati they're up to positive two seven goals they have scored and conceded five here's Poiser trying to advance that cut out quickly by Magati it's with daily now Martin Jackson Jackson gets it back and yes this passing is now slick Rabina Barrett left it well understood he was offside Miller Jackson skipping by two switch trying to swing it across but that was called by the captain Jordan Akin's for Roger Clark it's a corner kick for Magati the funny thing about it is I don't think Barrett was offside on that occasion I don't think he looked over at the assistant he wasn't interested at all so I actually think he might have been onside Barrett and decided to leave it though the corner kick into traffic came to the substitute in the end number 17 that's Kevoi Barnes here's Martin from distance still wasn't a very convincing hold by the keeper did just enough to keep it out it's a free kick the for Roger Clark Jero James there his first hole of the ball I was about to say I remember you calling his name previously James and already almost 60 minutes played and remember that this game is reduced to 80 minutes Martin beautifully spread it Barrett he finished it oh my no that's finished yeah in the back of the net it hit the back frame of the goal Lewis there get in here's a goal got it on his stride and fired and it was so clean an effort that not even the net in was rattled here's a second look Chris Taylor yeah look at how he cuts in though just to cut the angle very nice good finish Powell totally exposed could really do nothing about it Powell and you can see that's hitting that back pole along the base and he has the flips as well not as expansive but got it done so five goals to the good now for and Rachele and Luis is first of the season as well to go with his two assists Roger Clark getting ready to make a change Sean Paul Bruce comes on for Rodin McDaniel here's magnitude Blair but he's well off yeah by a meter or two he was certainly having a water break on the sideline or something I'm not sure why he even attempted to go after the ball it's hoping that the assistant was joining him in that water break yeah they dream it was something maybe you're still celebrating his goal Rachele and Luis Roger Clark at the bottom of zone e they will remain here after this result and now have conceded some 23 goals so far their goal difference at minus 21 and as I said this young team but a young team but some hard lessons being learned here never qualified before for the second round Roger Clark their best showing came back I think in 1999 actually when they were still known as Balaclava high I think they finished third or fourth in the group if memory serves me correct that was the closest they have come but yeah you'd like to think that on the coach Miller who has been around for such a long time and and done it with LeCovier made deep runs that maybe in a couple years we could be talking more about Roger Clark's performance yeah that goal from Rabina Barrett he's now brought off the field in comes at number 18 Kimmel Thomas a goal and assist and an assist for Barrett good performance very good and he said involved in the second goal as well here they come again that's a pure pace with the number 10 easy does it it's another one it's another one for him Rashane Lewis to on the afternoon six for Magity it's happening all for their favor here in Santa Cruz used superior pace had great balance and yeah unsure uncommitted nonchalant the goalkeeping there no mistake from the finish Rashane Lewis second of the afternoon good strength by Lewis does to shrug off the defenders as well cut in front of them to ensure that he was in full control and I love that extra second he took just before he hit it beyond Paul the composure he showed and his second of the season and the game this will be the third time already this season that Roger Clark have conceded six or more goals did lose six nil at the hands of BB Koch and 10-2 to Lakovia so lots of work for Errol Miller to do defensively he did concede it was a young team and Errol White I should say why did I say Errol Miller coach White he certainly seems up to the task and you know some coaches would have been upset and showing a frown has maintained an even face here they come again Magity great pace on the flank cutting in sends it across here's Jackson now Jackson rattling the crossbar but it was on the underside several goals for Magity his first of the afternoon his first for the season evil face Errol White but he knows his team is in dire straits good work in the wide areas cutting as well and yeah that's a good finish that is a good finish by Chadric Jackson nice step over there good strike with the right foot good in-step technique and you're much too much pace for Powell and Magity making this a bit embarrassing for Roger Clark now seven to the good you've been able to control the middle of the park and they come again Magity this is Barnes dispossessed but it comes to Jackson Jackson oh good save that from Powell equal to the task Powell where he has been unconvincing before that was good enough on that attempt here they come again Jackson Barnes Lewis blocked they come once more daily referee Makina Brown they're indicating that daily look at that strike once more from Jackson firm hand from Powell to ever the danger Hanson under lots of pressure throw into Roger Clark here they come now Matt Donald sends it across away and clear here they come again Martin the captain weeding through the middle of the field Lewis sends it across Martin on the end straight to the keeper it's a free kick the Magity right at the edge of the 18 yard box Russian Lewis standing behind this one has two goals already substitution made Hanson off before Roger Clark Monday on now the change being made Monday number 20 McLean Kimani McLean coming off and Sean Ferguson coming on and he's just giving younger players an opportunity Lewis certainly going for glory there that's a poll behind it but never got over the ball Lewis it was pretty obvious that one was going to go high just didn't get his straight right but yeah certainly Magity in full command all of the play happening in the half of Roger Clark and just about 10 minutes to go Roger Clark with a kick here they come again look at the pace look at the pace from Miller Powell really had to come out very strongly it seems as if the other players the defensive peer for Roger Clark they were moving in slow motion Martin to Jackson Jackson oh they're trying to do the what is fancy that it was Barnes it seemed the number 17 who came on well Thomas they were the miss kick daily he was battling to keep that one in his hands it seemed to be a hot potato that shot of moments to laugh Chris Taylor I have some interesting updates for you as well Dean Smith please Haydell Charlie Smith locked at Nillall in Manning Cup action Calabar leading Kemper down by four goals to two Calabar team this season certainly looking very good and a difficult group they're in KC also in that group leading by eight goes to Nill against Penwood there's Jackson inside and they're really getting more and more in discipline Magity yeah offside though for sure yes KC with an eight Nill lead over Penwood and Mona and St. Catherine locked at one apiece that's another very interesting game that Manning Cup one between Mona and St. Catherine probably the two high flying teams in that group here's Magity again have to reset now Magity Thomas to Jackson Jackson gets by two sends it across it was a shot actually that's a penalty handball was it yeah it was an easy call for the referee Jackson into the area just look at it here I'm away from the body that was hidden on target easy call so I wonder now who will take the penalty Miller already has two and Dante Blake there showed the yellow card for that so Miller was on two so was Lewis the substitute Kimmel Thomas or make that make that the captain Daniel Martin number 13 the one who's going to take this and it would be his first for the season now he actually has two already I mean his first penalty for the season well two goes behind his name though and two assists today steps up converts easily Daniel Martin the captain very confident player very confident player the number 13 Daniel Martin where's the captain's armband and just placed it beyond Powell almost wrong footed Powell as well who was leaning towards his right hand side was Powell couldn't get down in time third of the season for Daniel Martin he is the leading goalscorer for this Magity team and he has two assists as well to go with it he's being substituted just after that a bit of fanfare on his part in comes Romando Williams for him here's daily was queuing up for a shot that didn't come here they come again Jackson Chadwick Jackson since Kimmel Thomas has come on Jackson has been forced further upfield here they come again Magity unable to clear rather Clark and that miss attempt yeah Miller looking for his hat trick wasn't he I'm not sure you'll get a much easier one than that though Miller but has always falling away lots of smiles on that bench for Magity based on their performance it's a eight prong finish as it stands just about two minutes to go of course this game reduced to 80 minutes because of the interruptions so what is for you as well from the Dacosta Cup Rossi's lead from by a goal to nil fair and Fogarode locked at one a piece in another Clarendon Dabi and Mannins leading Grinch Hill by a goal to nil in the west Magity trying to go forward Shana Daly blocked yeah he's anxious to get on this score sheet is he Daly he has an assist already and I think his team is trying to queue him up quite a few players in line for a man of the match award yeah you would say Miller has two Roshane Lewis has two which he scored late in the second half as well as an assist Roshane Lewis and the captain Daniel Martin I like his play it was a controlling force in the middle of the park yeah was Daniel Martin the captain yeah to assist for Martin and then the penalty in the 77th minute I think he would probably come down to himself and Lewis Lewis has had a really good game as well I agreed you I like the calming presence of Martin in the middle and that's where they really come now Roger Clark can this be a moment of glory get it ready for the Sportsmax app moment brought to you by the Sportsmax app and it was this forward thrust by Lewis coming forward and the Jackson with the finish bang on target yeah really a grid strike got skipped by the man margin there was spoke of look at that finish that's the Sportsmax app moment brought to you by the Sportsmax app courtesy of that man Chadric Jackson I don't know there'll be much time for stoppages either normal time has elapsed already don't let that clock fool you the match reduced to 80 minutes and the word coming is that no time for stoppages everybody ready for that second game Monroe versus Saint Elizabeth technical and I'm sure Roger Clark would love to hear the whistle at any point now big win this for Magity Mckina brown close the whistle indicating the end of this encounter end of the the cluster cup Magity beating Roger Clark eight goals to nil courtesy of a couple of braces one by that man Rachele Lewis another by his partner in goal scoring brilliance Jackson and also Miller registering their first win off of the campaign so far Magity eight goals so that is the full-time score Magity eight Roger Clark nil this in the is a schoolboy football to cost the cup competition here in the home of champions those are the goal scores as you see them and Daniel Martin certainly the captain would have stood out as a man for the man of the match in both of minds yeah here the full match highlights started the goal scoring started in 17 minutes this ball played out wide Rabina Barrett blasted it by the goalkeeper that was his first and he fared home with a plum Rabina Barrett had the flips as well that was goal number one it would continue the 20th minute advancing was Barrett he pushed forward with great pace and the goalkeeper didn't come fell fortuitously to the post and yes Miller finished with fury as Wayne Miller opened his account in the 20th minute slid on the grass once more Miller getting that pass beating the goalkeeper and his near post that's goal number two goal number three would come number four rather would come in the 51st minute poor work there from the goalkeeper and Robinson Jereen Robinson was always a try and always fighting and finally he got his name on the score sheet the 60th minute another goal advancing was Lewis the number 10 and what a finish it was it was so clean it almost fooled me Chris Taylor yeah look there not even the net rattled it was a clean strike and he too had flicks the 64th minute another goal maggoty it was really one-way traffic what a run from this man Lewis again easy does it and this time the net rattled and that was his second of the afternoon in the 64th minute and they were in easy street maggoty high school the 66th minute he now proved provider Lewis look at this ball sent it across Martin skipped it and a Jackson a sportsmax at moment parted in with fury great strike that absolutely fantastic strike this was the moment there was a penalty Miller played it on handled and the captain martins stepped up and converted easily and that was the end of scoring all goals to maggoty here the full-time match statistics 36 shots for maggoty only one for roger Clark 16 of those in target none for roger Clark three false maggoty eight for roger Clark two yellow cards for roger Clark one for maggoty seven off sides nine corners all go into maggoty 88 percent of the possessions of the possession for maggoty and eight goals for the good we go to Gerard he's with the water man of the match thank you so much Dean I'm here with the water man of the match martin the number 13 man for maggoty let me just ask you this question quickly if you were to sum up your teams performance if you were to sum up your teams performance how would you say the game went before you go for you all today the game went good i am not the game went good both teams play both teams play really well and we just do what they have to do so thank you so much well done to you and hopefully your team can get more wins coming going on in this competition yes absolutely yeah that was a one of the match there martin and i was have a chat with coach white coach obviously not the result that you would have wanted however you did say at the start of the broadcast that it's all about rebuilding and learning how much do you think your team has learned today firstly congratulations to the other team we did not manage the game as well as we should especially in the 1v1 situation and therefore collectively we did not defend well either we're always you know behind the ball chasing but i think either is a learning curve for us and so you know i think they learned something from all right coach well we'll see you for the rest of the season thank you so much yeah well let's coach right there let's have a coach i'll chat now with coach Xavier from maggoty coach this is the result that you would have wanted now you get a win get your you move up to fourth place now join fourth place on points uh do you think now this is where the season starts and you can only get better from here yeah we're hoping that it gets better from here i think once the guys put in more work and work hard and quick on the field i think it should get a better result in the other games to come yeah what was the brightest moment for you with this game well i thought when you move the number 10 out wide and i thought we created the opportunities there and he came out and brought some life into the game first all right coach well congratulations and thank you so much yeah that's the end of this one maggoty eight little winners over roger clock high school at the centelismith technical high school sports complex in santa cruz in the part of centelismith this in the decosta cup that's the end of this match we stand by for another great fixture coming up inside this group in centelismith