Okay, so we've been talking about binary BCS codes, okay, and we saw how it comes to properties, we saw the priority check matrix description, we saw encoding, okay, so from here we went to the generator matrix, right, so that was quite important. Then we saw encoding, in particular we saw systematic encoding, we saw came up with a very simple way of doing systematic encoding, then what else did we see? Then we saw decode it, okay, so one thing that comes out from all of those is that both these can be implemented with practical constraints, okay, some things are very much implementable, both can be implemented, hopefully I convinced you of that the algorithm is not too hard, in today's PLSA it's quite easy to do it, you can just follow the algorithm and implement both, okay. So what I'm going to point out today is some important remarks connecting these binary BCS codes to other kind of codes called cyclic codes, so one thing I should say is what I've defined as binary BCS code is actually not, rigorously not the binary BCS code, it's a shortened version of it, okay, but this is good enough for practical reasons, I'll come back and I'll fix that and tell you what the actual binary BCS code is, what we saw is very closely related to it but it's not exactly that, okay, so before we proceed to that I want to point out about something called erasure, erasure and reverse decoding, okay. So this will influence particularly going towards future decoding algorithms which are much more complex than what we have today, you have to first understand erasures, if erasures are an important step in the muddy, so the idea of erasure is the following, okay, so you have, so you typically when you have an encoder, you have a message M which gets encoded into a code word C, so what I wrote down in that model so far is that to C you add an error vector which is binary and you get R, okay, so actually in a practical communication system that's not what will happen, okay, so something more crazy will happen, so one more level of sophistication is to say that the C gets converted into some modulation alphabet, then that goes through a noisy channel, say add it across a noise, okay, and then you decode that, okay, so that's a slightly more sophisticated way of doing the communication channel, so actually the receiver, okay, the receiver on the receiver side will not only get bits 0 or 1, it will actually get some more information that often about the grid, okay, so what you can do at a very, very simple level is that the receiver will receive three possible things, okay, it either receives 0 or 1 or some additional symbol which we will call an erasure, okay, so what does that symbol represent, it represents the fact that the receiver is, has really, is not really sure whether what was received was 0 or 1, okay, so it's very, very uncertain, so it simply erases that and puts an e-bar and conveys to the rest of the decoder that I really don't know what I received in this position, it could be a 0, it could be a 1, that's why I call it erasure, okay, so if you can use this information, you use this information and improve your decode, okay, so that's the idea and the binary erasure channel, okay, so that's called erasure and error model, erasure plus errors, okay, on the receiver side, you'll receive a vector r, okay, all right, so the way to model this r is basically r is now not a binary vector, it's not 0, 1 anymore, okay, it can have three different values, okay, so r is say r1, r2, rn, ri takes three different values, it could be 0, it could be 1 or it could be an erasure, okay, so several things can happen here, okay, so if you are transmitting a 0, the noise could have been very negligible that you got 0 itself, okay, if you transmit a 0, noise could have been so strong that you got a 1, okay, so transmit a 0, 0 and then you get some median noise, it's maybe a very strong or not very weak, so that the receiver can somehow figure out that I'm really confused, I don't know if this is 0 or a 1, at that point, we'd like to put this e, which we call as an erasure, okay, so if you want distinguishing from e, e and the error, but then maybe we'll call them epsilon, okay, so we'll say, so we'll say, okay, epsilon is a symbol that denotes that the receiver is totally consistent, okay, so the decoder will receive r1 through rn and each ri is already a 0, 1 or epsilon, okay, and there is a mapping between how 0, 1 goes to this, remember here this code word belongs to 0, 1, n, okay, so how does this happen, there is a probabilistic thing here, so input is 0 or a 1, the output would be 0 with probability, let's say, let's put some probability here, so let's say 1 minus p minus e, okay, and the probability e, it can go into this erasure mode, okay, epsilon to probability e, and the probability p, it can become 1, okay, so that's the model here, because I'll have a symmetric thing for 1, the probability 1 minus p minus e, it will remain as 1, the probability e will become this epsilon erasure, and the probability p, it will become c, so this is my errors and erasures model, okay, that's the idea, so this goes from the erasure, alright, so now the question is, how can the decoder use this information, so that's the idea, that's the point of it, the decoder should be able to use this information, the decoder can't use this information when it's useless, but it should be able to use it, okay, so the location of the erasure is kind of a hint to the decoder, it knows that in those places things are bad, okay, so let's first connect with some general theory, if there is no erasure, if the probability of erasure is 0, there is no erasure, then we know if d min is the premium distance, p erasure can be corrected, p is expected to be plus 1, that's when i equal to d erasure can be corrected, right, this d min, right, if the minimum distance is d, the code can correct, 2d plus 1 erasure, so this is under no erasure model, okay, no erasure, okay, if there are erasures, it turns out, if there are, let's say, let me have you speak here, if there are u erasure, what should we call it, let's say u erasure and v erasure, okay, writing erasure again, u erasure and v erasure are correctable, whenever 2 times u plus v plus 1 is less than or equal to, okay, so this is the erasure plus 1, okay, so how did we picture this, so we saw this big little space here which had all the code words, and we said any two code words are separated at least by a distance d, distance is greater than or equal to d, so if you make t errors and 2d plus 1 is greater than d, it will always be closest to the code word that you transmitted, so the decoder what should you do, you should do this nearest neighbor decoder, because you find the code word that is closest to you, and then you decide that, that is the transmitted code word, that will always correct v errors, okay, so now we have a problem where we have u errors and v erasures, okay, and in claiming there is a decoder for a code with minimum distance d, which can always correct u errors and v erasures as long as 2 u plus v plus 1 is less than or equal to d, is that clear, so that is the meaning of the statement, I think we did that way, so the nearest neighbor decoder, the maximum likelihood decoder can correct t errors as long as 2d plus 1 is less than or equal to d, that is something that we know already, okay, and now I have a situation where there are some errors, there are some erasures, okay, of course I do not know where the erasures are, I know where the erasures are, so what is the problem with the erasures, I do not know if it was 0 or 1, okay, so that is the problem, and I am claiming that if in the received block somebody tells you that there were u errors and v erasures, you can always correct it by designing a suitable decoder, how will I do it, any ideas, how can I go about doing this, okay, so I do not want a decoder explicitly, I just want some ideas on how to approach this problem, I mean I think there are some, see for instance we did not have an explicit decoder for the 2d plus 1 also, we just said there exists a decoder, all you have to do is go to the nearest code word, I mean it is not very implementable, it is some big maximum likelihood decoder, okay, but now I have errors, errors as well as erasures, so I cannot define distance and all that, from erasure I will do distance, I cannot do distance usually, so that is the question. If you convert an erasure to an error, some crazy things can happen, look at this condition, in this condition if u is 0, suppose u is 0, v can go as much as v minus 1, so if they keep converting errors to 0 and 1, I might go closer to other code words, I might get many versions giving me the correct answer, right, so I may not be able to control it that way, okay, okay, so that is the idea, so that is important, so what we do is we simply drop the bits that were erased from your code, so what is that operation called, it is puncturing, okay, so you puncture your code at the positions of the erasures, you know the position of the erasures, right, so you puncture your code in those positions, okay, now for the punctured code, what is the minimum distance? d minus v, okay, and I have u errors in the punctured code, as long as 2 u plus 1 is less than or equal to d minus v, I can correct the u errors, okay, so you go to the punctured code, dropping the erasures positions, do a maximum likelihood decoding, correct your u errors, and then what will you do? you are still not done, right, what will you do next, okay, so that is the first step, the first step is here, right, what do you do next? how do you know that there will be only one code word? okay, how do you know that there will be only one code word? it can be more than one code word, so she is saying, okay, so I bottomed up the errors in the non-erased part, there are no errors on that side, right, this is the procedure I told you, we will get rid of that, right, go to the punctured code, correct the u errors, the u is done, okay, now she is saying, for the erased part, you try every single possibility and see whether you get the code word, my point is how do you know that you will get only one code word, the answer is correct, you will get only one code word, how do you know? yeah, exactly, so because you have 2 u plus v plus 1 is less than or equal to d, you know that if there are 2 different code words that refer inside only that v, the minimum distance will be less than d, think about how you might want to show that, it's not really hard to show that, okay, get rid of the u also, you will have to have that, okay, another way to think about it is, okay, so v is definitely less than or equal to d minus 1, right, v is less than or equal to d minus 1, so you know those places so if you go to the parity check matrix, right, so you can solve for the missing positions, I'll show you how to solve for the missing positions, okay, so let me prove this statement to you guys, first step is what, puncture c, puncture the code to remove the erased positions, erased positions, right, the minimum distance of puncture code is what, minimum distance of puncture code greater than or equal to d minus v, okay, since 2u plus 1 is smaller than or equal to d minus v, u errors can be corrected in the punctured path, okay, you're okay, so u errors can be corrected in the punctured path, now the only problem is figuring out the erased path and I'll show you some, two different ways of doing it, the second step is to correct the errors, I'll show you a very simple way of doing it, the other way is to exhaustively try everything and argue that there'll be only one, I'll argue that there'll be only one in another way, okay, so we'll take the parity check matrix for the overall code, it'll have several columns, okay, so convenience I'll list the erased columns in the first position, it can be in any position and I'll list it in the first position, let's say the first column are erased, okay, so these are the erased path, so h1, h2, h3, whatever you know about me, okay, and then we'll have all the columns, okay, whatever you know about h times, so remember that this path I've corrected errors, I've already corrected, okay, so here instead of this new unknown path, let me put some variables, x1, x2 through xmu, so h times x1, x2 through xmu and the remaining path I know should be equal to 1, should be equal to 0, right, so that's a valid code, right, it should be equal to 0, so that gives me a set of equations in the unknown variables, right, the corrected path I'll multiply this guy, so I'll get one right answer, so I can get an ah equals b, the only thing I have to argue is ah has full column run, how do you know that ah has full column run? you know, it's less than or equal to b minus 1, so obviously that's full column run, so it's a lot of unique solution, I can solve it, it just starts with a solving a linear equation, so it's very easy, okay, you want to adjust it if I have to solve a linear equation, it's a fixed complexity, how do we get points? is that clear? did you see what I did? okay, so if you want, so let's say the corrected places are c mu plus 1 hat so I have to c and hat, okay, so for unknown ones, I'll call it x1, x2 through x2, okay, what do I know for sure? I know that h times x1 through xmu c mu plus 1 hat all the way down to c and hat should be equal to what? should be equal to 0, okay, so what do I do? I split this in terms of two parts, okay, so let me call this first these are columns, okay, so what I call this h1 through h2 are all columns, okay, so this sub matrix, let me call it some a, okay, so what I can do is I can split this multiplication as a times x1 through xmu equals what? the remaining part of it multiplied by c mu plus 1 through c and hat, I know all those bits, so this sum vector, so let's say sum, I don't know, sum vector, so let's say a1, a2, a2, a3, a4, okay, a and minus a, sum vector, okay, so if you know this vector, okay, you know this, these are unknown to you, okay, when can you uniquely solve it when the column rank of a equals mu, the column rank of a equals mu, you can uniquely have a solution for this, we can have only one solution, we cannot change it. For any a1, a3, a and minus k, we'll have a unique solution as long as the column rank here is equal to mu, okay, so how do you know that the column rank of a will equal to mu? That comes from the minimum distance, mu is less than or equal to t minus 1, so clearly column rank of a equals mu so that's the idea, mu or vi, right, we're saying v and v became mu, looks like that, is that okay? So that's the idea in claiming that as long as you have u errors and v erasures, 2 u plus v plus 1 is less than or equal to t, they can be corrected, okay, so this is the idea, so this is independent of dch codes or anything, this is a more generic idea, but there is an important idea here, the idea that is crucial here is by giving your receiver more information than just 0 or 1, okay, you can do something more, you can do something more, you can do something more here, so for instance if u is 0, you can correct up to d minus 1, errors, assuming that you know where the errors are coming, so if you give your receiver more information about what happened to each bit on the channel, you can do something more, so that's an important idea which is exploited in today's communication systems, in today's communication systems for each transmitted bit 0, we'll at least give 6 or 7 bits to the receiver, so it's not just 0, 1 and epsilon, it is 6 or 7 bits of, what is called soft information, so we'll do a very granular detail, so we'll say 0 can be received in any of these forms, so you just basically provide much more granularity to your received value, and that can be very much used by the receiver and you can really get very good performance, so this is just like a baby step towards that, but anyway, so this principle is used very much in today's communication, so if you're doing the next course, you'll see how that is really exploited, so if you give more information to the decoder on every bit, then it is much more useful for decode, so let's come back to BCH codes, so suppose I say in a BCH code, I'm decoding BCH code, I know how to correct a 2T errors, as long as 2T plus 1 is less than 2D, I can do that, I have a very efficient algorithm for it, now suppose I say, I want to use the bounded distance decoder, but now I have URS and D erasure, I want to use the bounded distance decoder, I don't want to do, not some likelihood decoding in the puncture coordinate, I don't want to do any of that stuff, I want to use the bounded distance DCS decoder, because I know it's very easy to do, otherwise it becomes complicated, so the question now is, how do you implement errors and erasures decoding in DCN codes, in the bounded distance decoder for DCS code, how do you interpret it, so that's the idea, so let me talk about that briefly, so in DCS decoding, suppose you have URS and D erasure, and let's say 2B plus B plus 1 is less than or equal to B, so how to adapt a 2B bounded distance decoder, so that's the question, I want to prove, any ideas, how can I do this, how will you do it, let's say the problem is complete, you can't drop, how do you compute the syndromes, you can't compute syndromes, you can't drop, I'm not dropping, you see, when I say drop, I'm puncturing, I'm not making it 0, so it's a completely different thing, so yeah, you have to answer, how will you compute syndromes, let me ask you, so in the previous case, you go to those positions, you go to a punctured code, and do maximum likelihood decoding at a punctured code, it's everything you can do here, except that it's only theoretical, you can't implement it, so you're not going to be able to do any easy decoding with the punctured code, it will do maximum likelihood decoding, same method, no problem, yes, that is exactly the problem I'm posing to you, so here we have some dots in the app, what do you do, you don't have the received values, how do you do it in the first step, because with the syndromes, you can't do it, without the syndromes, you can't do the algebra decoding, so that's my question to you, how will you do it, there is a very smart and slick method, this will be available, that's why I'm presenting it to you, so you'll see it's a very slick method, so when I describe it, you'll see that it's a very clever method, it's not a very fundamental, or conceptual method, it's just a clever method, so what is that clever method, so I'll give you the first step, is to rewrite this in this form, 2 plus b by 2, it's less than or equal to t, let's say d is odd, so that t is even odd, that's the first step, so u plus b by 2, I know, is less than or equal to the order correcting capability of my code, can you use this now, come up with a clever idea, what can I do with erasure, suppose I don't know the erasure, what's the first step that you are always suggesting, what do you try, are you trying, what can we try, let's just give me one simple thing that you can put the erasure to, start off with, all 0, let's say I try all 0, right, u plus b by 2 is less than or equal to t, I know that, so b by 2 is less than or equal to t, right all 0, let's say something happened, then is there anything else I can try, all 1, so in fact that is enough, so you put, erase position equal to all 0, you do one decoding, then you put erase position equal to all 1, you do another decoding, it turns out at the output of these two decodings, you can combine the output of decoding, these two decodings smartly to get the actual decoder, as long as u plus b by 2 is less than or equal to t, okay, so you think about why that works, it's not very hard, just a clever method, so what you do is, you put, here's the method, you put erase plus erase plus equals 0 and do down the distance decoding, do the algebraic decoding, okay, now once you put it is equal to 0, you can do the decoding, so it's just a question of computing the syndrome and running the decoding, you do algebraic decoding, okay, and then what we do, next step is, do the first step, second step is, you put erase plus equals 1, all of them equal to 1, then again repeat the algebraic decoding, okay, so you might say that okay, only one will succeed, whatever succeeds can be the output of the decoder, so you can think about why it is true, it's a very slick little idea, you will see that, if one succeeds, it means, well, other will definitely fail, because the number of others within that would exceed where, exceed t, okay, because b by 2, if you put all 0, b by 2 has to be, because it's very nicely, see, if you put all 0, the actual error vector will have some weight, that weight is either less than b by 2 or greater than b by 2, it can't be both, so if it's less than b by 2, then when you put all 1s, the number of errors within greater than b by 2, so that's the way you control it, so you have to prove it, I'm not going to prove it, but this is the idea, so what's better than b by 2? no, no, no, so when you put all errors bits into the 0, how many errors will actually be there in the new, in the new received word? okay, the received word, some portions are erased, you put all of them as 0, how many errors will there be in the new received word? it will be u plus what? weight of the error vector in that part, okay, when you put all 1s, what will it be? it will be u plus b minus weight of the error vector in that part, okay, one of these two things has to be smaller than b by 2, the other will be definitely greater than b by 2, okay, so it will work out better, okay, so if both succeed, then you have a problem, both succeed what happens? you pick the one with the number of errors, so that's it, okay, if either one succeeds, you pick one, depending on b is high enough, then only other one will succeed, if b is very, very low, b is 0 for instance, when u is less than t, then both will succeed, both may not succeed, so you have to think about what will happen, b is very small, yeah, b is 0, both will succeed, right, so both succeed, then you have to pick the one that gave you the lower number of errors, okay, both give you the same code, both will do it at that time, it's not a problem, okay, that will be the correct answer, okay, so you just combine the output of these two decoders in a smart way, you get the answer, okay, and prove that it works, okay, so that's about BCH decoding with errors, and now we're going to move to the other part that I was talking about, this is the BCH code stuff, okay, so like I said, the codes that I designed for you are not really binary BCH codes, they are sharpened versions of the binary BCH codes, the actual binary BCH codes are found in this class of what I call cyclic codes, and I'll tell you what they are, okay, so first let's look at a special kind of BCH codes, let's look at BCH codes, okay, for which n equals the order of beta, okay, so this is the special condition that I'm going to import, okay, so far what did we want either of beta to be greater than or equal to m, so now I'm going to say I want it to be equal to m, okay, so of course now n has to be odd, okay, so if I'm going to look at, if I'm going to want beta to be in 0 to bar n, n has to be odd, only then it will work, okay, so n is even, there's no way order of any element in g of 2 bar m is going to be even, okay, why can't it be even? It has to so the order of any element has to divide something, okay, so we have to divide 2 bar m minus 1, clearly an even number is not going to divide 2 bar m minus 1, so there's no way order can be even, so n will have to be odd, okay, so let's take n odd, and then we will ask for n, beta such that n equals order of beta, okay, then there is a very, very curious property that is satisfied by the four words of the quote, okay, so let's go back to the barbecue check matrix, what is that barbecue check matrix? 1 beta beta square all the way to beta bar n minus 1, 1 beta square beta square all the way to beta square all the way to beta square, it's still done, okay, so we're going to 1, we've got 2p, for some reason it will be equal to 2p plus 1 all the way to the standard, now we're going to 2p plus 2p plus beta, so this is what I'll just check matrix, suppose I tell you c which is, it says c0, c1, c2, all the way around the cn minus 1 belongs to the code, what does it mean? h times c transpose equals 0, okay, this implies, and we reply by for instance, that a cyclic shift of c, what is a cyclic shift of c, I move to the right, and then bring the last one back, the cyclic shift cn minus 1, c0, c1, cn minus 2 is also in the okay, so where is that to? well, it's not question of swapping, how will I show that if c belongs to the c0, cn minus 1 belongs to the code, the cyclic shift is also in the code, what is the way to show a code word belongs to the code, an actor belongs to the code, you have to check the distance between the cn cc transpose, okay, what will be the cn c transpose and cc you take the cn c transpose, you multiply by the beta, what will you get, look at the first equation, 1 plus, so the first equation multiply by beta, okay, what will be the first equation, 1 c0 plus beta times c1, with beta squared c2, plus beta power n minus 1 c n minus 1 equals 0 you multiply the beta what will you get yeah so it will become it will give you this the equation corresponding to the first first equation corresponding to the second vector and that is also 0 what will I give you the second one? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . codes for which you take a code word and do a cyclic shift, you again get another code word called cyclic. Linear codes, there to be linear, but also you can think of non-linear cyclic code, but you should know everything about it. So, linear codes that a cyclic right shift, I can say right shift without any ambiguity, left shift is any number of right shift put together. So, cyclic right shift also gives you another code word, it is called a cyclic shift. So, this makes the code cyclic. I mean this will not happen is n is not equal to order of beta. If order of beta is cyclic greater than n, you have a gap, so you can't do that. So, only when n equals order of beta, I can do it. So, this leads us to this area of cyclic codes. So, these BCH codes are special cases of cyclic codes. So, in fact, suppose order of beta greater than n, let me show you what happens for n order of beta greater than n. Suppose if your order of beta is strictly greater than n. So, what you should do is, you should define a BCH code with length equals order of beta. So, you define a BCH code with length equals order of beta. That code will be cyclic. And then how do you go from that code to this gate? How can you go to this BCH code from that code? You shorten it. So, you have to set some of those things to 0. You cannot puncture. Puncturing is different from shortening, right? Puncturing is drop the bits. Here you cannot drop it. You have to set those bits to 0. Only then the clarity set matrix will drop out. It will become the shorter one. So, you can shorten it to go to that. So, all BCH codes are shortened versions of cyclic codes. So, cyclic codes are in a way more fundamental than the BCH codes. So, you might say, I can say cyclic, but it is very common to say cyclic. Everybody says cyclic. So, I am also cyclic. If you want, you can call it cyclic. So, apparently I am called in English the right way to pronounce, for instance, by cycle is to say by cyclic lesson. Because apparently maybe it is a lesson in pronunciation. So, we say cyclic just to keep it common with our algorithm. So, it makes sense to look at cyclic codes fundamentally. So, what codes can be cyclic? So, what is the definition of cyclic codes? Linear codes, closed under cyclic, right? So, what do I mean by closed under? So, this right shift can also be replaced by left. So, but they are not equivalent. Closed under means I have a code, I do an operation, I do not leave the code. So, if that is the thing, then that operation is said to be closed under that. I mean the set code is said to be closed under that operation. So, here the code is closed under cyclic, right? So, such codes are interesting fundamentally from various reasons. They have a very nice rich algebraic structure. And interestingly, VCH codes are special cases of it. A shortened version of it in general. So, maybe we should look at it. So, we will see it real quick in a couple of classes. It is not too crucial. Like we saw I mean the cyclic property was not really used anywhere. We never used it. We only used one property which also is true for the cyclic one. We used this notion of a generator polynomial, right? So, that is quite, that we come from the cyclic code in a very nice way. So, that is a very nice characterization except for that nothing else we used. So, let us see some theory. The first thing is we need a nice mathematical notation for our conceptual understanding of the cyclic right shift operation. The cyclic right shift we know what it does, but how do we describe it in algebraic structure? So, this is the way to do it. We will first define some ring which we will call Rn. Ring Rn is basically, we will only do binary. So, we will only do binary. So, before I do that, we will only do binary cyclic codes. So, binary cyclic codes you will usually set n to be odd. So, just fix these things. We are only dealing with binary codes and n odd. So, this ring Rn is basically polynomial as did we less than or equal to n minus 1 with binary coefficients. So, this is a ring Rn. Addition is trivial, right? Addition is very, very trivial. If you add two polynomials, you will get another polynomial in the ring. I have to define multiplication. Why should I define multiplication? So, I do a polynomial multiplication. What can happen? Can this be a ring? No. I mean, you will go outside of this set, right? So, you take two polynomials of degree n minus 1. You multiply. You will get a polynomial of degree x bar 2 and minus 1. I mean, there is no way that is going to be in this set. So, what can I do is multiplication is 1 modulo x bar n minus 1, okay? All right? So, I could have picked any other polynomial of degree n. I will pick this x bar n minus 1. Well, minus 1 is the same as plus 1. So, I will say x bar n plus 1. Because I am dealing with just binary stuff. Everything is gf2. So, I will say I will do multiplication modulo x bar n plus 1. Okay? The reason why I am saying that is if somebody tells me that I have cfx, which is c0 plus c1x plus so on to cn minus 1 x bar n minus 1, it turns out this definitely belongs to Rn, right? No problem. If I multiply cfx by something in Rn, I will get a cyclic ratio. Okay? What is that thing I should multiply by? By x. By x. So, multiply by x. So, x times cfx equals what? cn minus 1 plus u0x plus c1x bar all the way down to cn minus 2. So, in this ring that I have, multiplication by x nicely captures cyclic ratio of operation. So, what do we do for multiplication of cyclic ratio by 2 positions? x square by 3 m. Okay? I can even multiply by x inverse if I like. Honestly, I define x inverse suitably. I can do that. Okay? Do a left shift. Okay? I can do that. I define. I will do my ring. x bar n is equal to 1. So, x bar x inverse is the same as x bar n minus 1. I can define that. Okay? So, I can do a shifting very nicely in this ring. Okay? So, that's the first thing. Okay? So, we will interpret cyclic codes. Okay? Cyclic codes are linear codes. Codes are also subsets of this ring. Okay? So, codes. So, this ring is also a vector space. Okay? So, I can also think of the codes as a subspace of this ring. Okay? So, I can do a loose terminology, mixed up terminology. Then, a cyclic code. If I define a cyclic code of length n, somebody gives me a cyclic code of length n. Every code word of that cyclic code belongs to R. Okay? So, I can think of my entire code, cyclic code, as coming from R. Okay? It comes from R. Every code word comes from R. Okay? Okay? So, using this ring, I'm going to study the cyclic codes. Okay? So, we will not use too many properties of this ring. I'm just defining it for completeness. Then, we will see what we do there. Okay? So, now, the next thing that is defined. Okay? So, let's go back. So, vector spaces, we have subspaces. For groups, we have subgroups. For rings, there is something called an ideal. Okay? So, that is what is contained in a ring, and it utters the properties of the ring. That's called an ideal. Okay? So, subspace, we define a vector space. Okay? So, we define the linear code. We add a vector space. We define a subspace for it. And, for group, we add a subgroup. Subgroup and subspaces satisfy only the vector space properties. A group here, only one operation. In the ring, you have two operations. Okay? So, if you have a, you can have a subring, but that's not so interesting. An ideal is much more interesting. So, what's an ideal? If we define an ideal as a ring. Okay? So, I'll define it with respect to Rn. We have to strictly define it with lots of intricacies here. So, let's just define ideal as Rn. Okay? A subset of Rn. Okay? Okay? It's an ideal. Okay? So, there are two conditions that need to be sub-spaced. So, let me write down both of them. Let me write down both of them. The first condition will be very easy. So, we'll do C. If we have A-effects, B-effects in I, then R-effects plus B-effects should also be in... That's a very simple condition. It's just a standard addition. So, it's like a subgroup of the additive groups. If you view Rn as an additive group, it's a subgroup. If you view Rn as a vector space, it's a sub-space. Okay? So, that's the first thing. Second property is if you have an A-effects in I and the B-effects in Rn. So, remember that. If I have an A-effects in I, I have a B-effects in Rn. Then B-effects times A-effects should actually belong to I. So, this sounds like a very, very strong requirement. So, I'll motivate this a little bit later. This is what's needed. Is that okay? It's a bit strong. You might... First time you look at it, maybe I made a mistake or something you might say, but no, this is correct. A-effects belongs to the ideal. And B-effects is any other polynomial in the entire ring. So, multiply B-effects to the A-effects. That's what I go to. So, come back to the ideal. I should not bore it. Okay? So, for instance, the integer ring is something that is very common with them. So, if I take all the multiples of, say, 3. Okay? So, if I look at the subgroup of the integers, which consists of all the multiples of 3, that will form an ideal. Okay? Because if I add any two multiples of 3, I get a multiple of 3. So, take any other integer and multiply it by a multiple of 3. What do I get? I get another multiple of 3. Okay? So, in a way, these ideals are multiples of... Ideals and integers are multiples of a number. Okay? There is an interesting converse to this statement. Okay? If somebody tells you that there is an abstract ideal for an integer, you can show that there is a number such that that ideal equals all multiples of that number. Okay? There are no other ideals other than multiples of numbers and integers. Okay? So, it's a very intriguing aspect. There is a very simple statement which is true for R and N also. Okay? So, that's what we will prove. Okay? Now, let me just unpack this for you. So, what is bfx times afx belonging to i? Now, the bfx is some arbitrary polynomial from R and N. What form does it have? b0 plus b1x plus b2x plus so on till bn minus 1, x bar n minus 1. Okay? So, somebody tells me that i is an ideal. Clearly, that implies that afx and i, okay, x times afx will be what? What? x times afx will be an i. That's no problem. x bar times afx will also be an i. So, any power of x times i afx is also an i. Okay? So, instead of requiring that bfx times afx is an i for every R and N, what can I simply require? Like, x times afx is an i. That's enough. Okay? Okay? If x times afx is an i, what will happen? x bar times, so that's enough. Okay? So, that's what I'm trying to say. So, even this power i would only, okay? If I just require x times afx is an i for every afx, that, in fact, implies this condition. Why is that? You take an arbitrary dfx and multiply by afx. What do you get? You'll get an x times afx here, which is in the ideal. You'll get afx here. Then you take x times afx and multiply by x again. You'll get x bar times afx, which is again an i. Like this, all of these days are in i. And then you multiply it in scalars and add them up. What will happen? You'll never realize. Okay? So, maybe I should be capital S and say I'm in binary. Well, good. So, if I'm not binary, I have to add some scalars here and say... Okay? Okay? So, the second requirement in Rn can be replaced by just this requirement. Okay? If I have afx and i, x times afx belongs to i. I introduce cyclic codes. Now, what's the connection between binary, linear, cyclic codes and ideals of Rn? They are one and the same. So, once you see this, you can easily see that binary, linear, cyclic codes or binary, cyclic codes are ideals of Rn. Okay? So, this is the first and interesting result as far as cyclic codes are concerned. So, once you understand ideals of Rn, you understood all possible binary cyclic codes. There can be no other technical cyclic code other than... Okay? So, how do you think of ideals of Rn? Okay? So, it turns out, like I said, the crucial thing is this idea of multiples. Okay? So, in fact, in Rn also, that is true. Suppose I take... So, here's an example of an ideal. An example of ideal. Suppose I took some gfx from Rn. Okay? And then I can generate an ideal with it. How do I generate an ideal with it? Define id to be set of all multiples of gfx. Okay? So, you take all possible overfax times gfx. Then, overfax is in Rn. Okay? This will be an ideal. Okay? How do you assure that this is an ideal? Okay? So, you fix some gfx. And then you do this. This will be an ideal. How do you assure that this is an ideal? Okay? You take any two sum... Sum it up. We'll again get an element in the ideal. You take gfx and gfx multiplied by some other polynomial in Rn. We'll again get an element of this ideal. Okay? Because all possible multiples will be included. Okay? Multiplication is associated. It's a problem. We can do all the simple properties and see that this will be an ideal. Okay? It turns out, again, interestingly, a converse kind of statement is true for such a statement. Every ideal in Rn looks like this. So, for every ideal in Rn, somebody comes up with an abstract ideal in Rn. Somebody gives you an ideal in Rn. You can ask him to provide a gfx or you can compute the gfx such that the ideal will be equal to this. So, these kinds of ideals have a name. These are called principal ideals. Okay? The reason is they are generated by a single element. Single element from Rn generates the entire ideal. Such ideals are called principal ideals. The statement that is true for Rn is every ideal in Rn is principal. Okay? Every ideal in the integers is principal ideal. It is just one way to generate the whole thing. Likewise, Rn is also what's called the principal ideal domain. So, it has every ideal being principal. Okay? So, that's the first thing we'll prove. Okay? So, that's the first thing we're going to prove. Okay? So, let's try to prove that statement. So, here's the result. Okay? Suppose you have a cyclic code C. Somebody gives you a cyclic code of length n. Remember, n is what we are. C is the binary code, all that we'll assume. Okay? All that we'll assume. Okay? So, clearly, this is an ideal of Rn. Okay, that we know. So, instead of calling ideal of Rn, I'm going to say cyclic code. Both of them are equal. Nothing good. The first result is that it exists. Okay? So, let me go step by step. The first result that I want is that g of x be, be, be, be, be, be, be. Non-zero polynomial. Non-zero element. Okay? So, let me just say non-zero polynomial. Polynomial in C with minimal degree. Okay? Okay? So, zero polynomial really doesn't have a degree. Okay? So, I don't have to say non-zero polynomial. Once I said the polynomial has a degree, it was non-zero. Okay? So, anyway. So, suppose I say g of x is a polynomial in C in the cyclic code with minimal degree. The degree is minimal. Okay? Then g of x is minimal. Okay? It's very easy to prove it. Okay? So, I do a prove this. If I say the non-zero polynomial in the cyclic code with minimal degree has to be unique. Why is that? How do you prove it? Yeah, if there are two of them, you subtract the two. Again, you are in the cyclic code. But that will have a degree strictly lesser than either of them. So, it cannot be proved. Okay? So, it's a very easy to prove this, for example. The next statement is C itself. So, there's a notation here. For such principal ideals, these are denoted g of x. Like this. Okay? So, what do I mean by this? Basically, this way is denoted as the ideal generated by g of x. It's just a short-hand notation to denote principal ideas. Okay? So, it turns out C equals two of x. Okay? Is it okay? Fine. So, how do you prove this? Okay, I prove this. Any ideas for proving it? There's a non-zero polynomial. Doesn't that take any code for that? Yeah. So, how do you use the division algorithm? Okay? So, you take, let C of x pull out to C. Okay? Then you go right. C of x right. C of x. Okay? Where? Where will you divide this? So, don't divide in Rn. Okay? Rn is a crazy thing to divide in. So, you have to divide this in g of 2 x. Okay? So, this is a regular division. Okay? So, then we'll get a quotient and then a micro. Okay? So, now, if I do, if I push this to this side, what will happen? I get C of x, put C of x to C. And then I get it. C of x plus g of x. What does this belong to? If I do, say, model x minus 1, 5 minus 1. So, I don't have to do it because C of x has a degree, less than or equal to n minus 1, right? So, if I divide by g of x, which also has a degree, less than or equal to n minus 1, I will not exceed degrees in any way. Okay? So, this will be fine. Both of these case belong to C. Why does this also belong to C? C of x belongs to C. Any multiple of g of x also belongs to C. Both of them belong to C. So, what about the sum? Both of them also belong to C. Okay? Now, I know a degree of r of x is what? Strictly less than degree of g of x. And this will not be the contradiction only if r of x is equal to 0. Okay? So, this, why? Because degree of g of x is minimal, right? You see that? Okay? See, I have a model. Suppose r of x is not 0, what will happen? R of x has a degree strictly less than the degree of g of x. And r of x also belongs to C. Okay? So, that is the contradiction of the minimality of the degree of g of x. So, that implies r of x is 0. Okay? So, every c of x in C has to be a multiple of g of x. Because we are going to various other ways of showing it. Please go and pick this one. Is that okay? I think about that. So, I mean, I will take just two more minutes and give you the last property which is again interesting. The third property is g of x provides x star n minus 1 in g of x. So, this polynomial that we have, which was my polynomial at minimal degree, minimal degree in basically code. Okay? Not only generates basically code, it also has to derive x star n plus 1. How will you prove it? Whenever something has to be derived, you always use the remainder. Take x star n minus 1 derived by g of x. Okay? You will get q of x times g of x plus r of x. Degree of r of x is strictly less than degree of g of x. How will you do not even x star n plus 1? What will happen? x star n plus 1 will go away. So, you see that q of x times g of x belongs to the ideal, belongs to the cyclic code. Okay? And it is equal to r of x. And r of x is degree strictly less than what g of x is supposed to be. That is the contradiction. So, that cannot happen. So, r of x has to be 0. Okay? So, this polynomial of minimal degree in a cyclic code is a power cell polynomial. Once you get a handle on it, you know everything about the cyclic code. Okay? So, that polynomial is called the generator polynomial of the cyclic code. Okay? All right? So, the generator polynomial for the BCH code that we had is also the generator polynomial of the cyclic BCH code when n equals order of beta. If n is lesser than order of beta, then you have to go to that cyclic code which is the full cyclic code. So, that it will be the generator polynomial. Okay? So, that is the connection. Okay? So, BCH code, the ocean you defined only when n equals order of beta. If people will not define BCH code, so n less than order of beta. If you want n less than order of beta, you have to shatter me. But I think they are just saying BCH code mostly for it. So, hopefully that is clear. Okay? So, we will stop here for today. If the people who want to do projects have some time now, we can have a quick chat. Okay?


So please tell us a little bit about the early life, the early gay life you experienced in London. What sorts of places and things did you experience? Okay, well, again I'll have to backtrack slightly. The first London experiences were trying to find anything sexy anywhere. Okay. So as a very gauche 17, 18 year old I used to go trolling around Soho just to see what I could find. Just being the Soho Square and all these sleazy side streets with porno magazines, shops and stuff was really quite exciting. In those days you could actually look at them. Later on it turned out they've shrink wrapped them all so you couldn't see them. So that was where it started and then at some stage, I can't remember exactly when, I thought I'll try a strip club. So coming off the street and there was this woman sitting at the bottom of these middle aged women sitting at the bottom took me money. And then going through another door and they wanted some more money.  I said I'm not paying twice. I haven't got no money. Oh well go on then because they could see us quite young. And that was a straight strip to see.  And then I found the name of a cinema. It was a Bijou. Around the back end of Leicester Square somewhere I believe now. And in that cinema not much went on in the cinema but I did notice a lot of people kept disappearing off to the toilets. So I thought well I probably need to go and look. I might even need a wee. So that was quite interesting. But that was, well I was a young bloke then, that was more older bloke. In fact I got caught on to that by the first bloke that moved in here with me. That was that we bought the house with because he was a Bijou fanatic. Every time we went down to London we'd be off around the Bijou. And then having got round the Soho and also I used to buy or get all the magazines I could find.  They'd had personal ads in which in those days were couched in careful terms. Like professional solicitor type would like to share his briefs with you and this sort of thing. Stuff that's all in your window. So I started placing one or two ads and answering one or two ads. And then I'd meet quite a lot of people in their houses or wherever. Some bloke did try to pick me up once just near Piccadilly Square. And I was quite happy to go with him but I hadn't got anywhere to go. I didn't have any hotels because I wasn't a rent boy. So that faded away, that was a big disappointment because he looked quite well hung. And clubs as such, I never did clubs a lot because well I suppose his studies got in the way a bit. And also I was never a great one for really late nights. But however what I did do, I gained through, see actually I got to know of parties that were going on. So I'd be off down to various London party venues, more so than clubs.  And it was a good comfortable place to chat to people. In some ways they were better than they are now. Because now they're, if you like, fashionable for the straight world. So it's changed the whole sort of look and feel on this change. And he made no sense, he liked them as well. But declined. Saunas, that's the next thing I'm going to talk about because saunas. So again, went to saunas more often. They were good. Used to enjoy saunas. So that's really what the early days in London were like. This would be, let me think what years would be. Well actually, probably not that early by the standards you're hoping for. But that would be around early to late 70s. It was most of my time down London. By the time we got to late 70s I was much more involved with my mate. And so I feel like the scene we didn't get in on as much. Which I did do most of the free time. Certainly before he moved up here anyway. The two years before he moved up. All weekends were his or mine. Just doing trips out and stuff. It was definitely improving. We did go to one or two of the clubs. Heaven and something else I can't remember. We then found out about SM Gays, I think. I'm not sure when they started. But go to some of their events. That was it really, I think. The world's greatest clubber. Because they do like the music. They stop playing rock and roll by that time. Do you even know where to find the publications that then led you to the places around London? Okay, well as I'd said. One of the CHE groups. I got to know people in then. The CHE groups and friends. They would know where pubs were. Because obviously there's some older men as well.  So you could go to the pubs. And a lot of the pubs stopped free of charge. Gay literature of the time.  One or two other sort of more contact type magazines. And you just picked them up on your way out. And that would sort of tell you where other places were. And then you could use that as a sort of chain reaction go from place. And then that would tell you where saunas were and things. And when you get to saunas you chat to people and they'll tell you where the other ones are. So it's just sort of grapevine, if you like. As much as anything. I think that's the main way it worked. I didn't have the internet in those days, of course. So it really had to be that sort of method. And of course if you met people at parties and got on then you could change details. And then you meet other people at other parties and carry on like that. The mid-1970s, which is the time you're depicting. Historically has been viewed as a very sort of swinging time in soho that soho was very free and liberal and lots was going on. How did you see the soho area? Well I did see it quite like that to be honest. But as much as anything I think that was probably optimism of youth because when you're young you see everything optimistically pretty much. And everything seems exciting. And of course the more you discover something new the more exciting that tends to feel. And it reinforces the idea that this is a good area. I mean I used to go in and around the soho pumps quite a bit. The main problem was not so much finding the venues but really connecting with people sufficiently to see them again several times start possibly relationships and things else was if you like a slightly harder aspect of life. But that's the same for Gallifogue these days to be honest. So I'm sure even with the internet I feel like friends on the internet but having friends that actually exist and that you get to know and trust that's different. So it did feel good and the laws had moved along a bit and the press was not quite as horrible. So it did feel better than it did in the 60s and of course before the 60s I was too young for anything about scenes I was much more into dinky toys and things. SM Gaze came up a bit ago in our discussion. Tell me how you accessed that and what that afforded you. Well when I said earlier about how I felt when I first got into the day of the tour which was just wonderful because I was with people on one and I always had liked the exploration of pain for pleasure. I mean it started as a self-interest that's the way I could put it which went quite away and then of course naturally after a while you want to meet some other people that like playing about like this and I do consider it in terms of playing about I don't consider it in any of the terms that would come on too late to use by the media and the courts and the police and so on but for me it was exploring as I said I used to have a quote pain is pleasure and that's how I used to see it because it doesn't appreciate you in door things it gives you a buzz so I'm in the modern equivalent of PNPs I can't say P is good honestly I think I'd rather just carry on with people than exploring pain so that's why I liked it and I wanted to find other people to try it with and on me I mean I was definitely a switcher I tried either way round if people wanted to try something and I didn't think it was going to be horribly dangerous there's only once I actually felt unsafe was I met one guy he was actually a hospital nurse trainee nurse or something I went out to his nurse's quarters once and he tied me up this was in a sort of bondage bit of a thing and then his eyes went sort of glazed I can't, it's hard to explain but he was just almost not there he'd like someone to click to switch and suddenly I didn't feel safe and I spent the next quarter of an hour talking him out of continuing and could he sort of just untie me and I did actually get a letter from him I left then, the following day I got a letter so he was sorry it's just a problem he had with it sort of thinking because there are dangerous people I'm not saying he had any ill intent as such but it just felt totally uncomfortable whereas I always wanted to be comfortable with the people I'm with you get more out of it if the two of you are feeling relaxed and comfortable so we explored the gay world SM Gays I can't remember where I found out about that but as soon as I heard it existed oh I know, it was some friends of ours that we'd met through our transport interests that's right, they belonged on the transport group that was a gay group, still going I believe what group is that? it's a group of gay folk who are interested in well it says transports but essentially they're trains and trams whereas to me transport is trains, trams, aeroplanes, ships sort of all transport and they also there are what you'd call a few sort of geek timetablers and charming the roots and all this sort of but that's them if that's what they want to do good luck to them we used to go to that, they used to it was much like a sort of transport version of CHE you'd meet round people's houses there'd be slideshow presentations you'd meet round people's houses and there would be slideshow presentations and talks and so on so that was there and then obviously with some of the members you'd get to know them a bit but all your fans see them a bit, let's be fair so you start chatting them up and in particular I chatted up a lot I used to go round and see him quite a bit and he knew someone else in London and it's going down to him that I found about the SM Gay thing and this they used to meet I think it was under Vauxhall Bridge it's one of these sort of arches type places and I'd go along to that and that was nice because that was basically SM related folk so you'd have lots of leather gear guys people would be bonded to have some games equipment you know cages, slings all the usual sort of gear nothing, they didn't have for instance lots of sex toys because I think infection risk would be too high in something like a club which was basically a club but certainly you could stick someone up in a swing or tie up to a cross or whatever you wanted to do plus chatting and all the rest of it so it was a nice atmosphere and that of course is a little bit like a sauna except you haven't got to go through the rig and I would be explaining what you like because the very fact that you're in this club that's SM based means they know fair enough that was of course, I've forgotten this that was the days of hanky codes I suspect your followers are familiar with hanky codes there used to be a whole list of them must have been awful, been colour blind but that was quite exciting as well and I used to have a variety of hankies depending on how I felt such as red pale blue which I think was wendlin I'm sorry I mean we really rubbish these days but I had a variety of hankies shall we say and they used to pop me in left and right pockets according to choice and so on the only thing I wouldn't ever do was any drugs apart from poppers cannabis was alright I didn't smoke, I've never smoked but I'm not averse to parties that had cannabis because it always used to have a nice sort of sweet smell to the rum and you'd drift off into a sort of, not a haze exactly it's not really high it's a sort of very relaxed state with cannabis and poppers and things well no, poppers isn't up and cannabis isn't much relaxed so that was alright, I didn't mind any of that but I wouldn't ever try LSD and all these things not for any moral reason because I was too much of a coward I wasn't sure that I might enough try and jump out an upstairs window to fly and have people in apprenticeships and aircraft industry and I'd already discovered that no matter how much I've flapped my arms I can't get off the ground and I carry on doing that well into my 60s I had to give up after a bust my shoulder one day so that was it, so that's where I found SM Gays and after that it was like as before you just get to know people and other people and then you'd have different groups and maybe little orgies I started doing they used to do orgies at this place as well definitively sex nice this was still against the law at that time because it was still counted as the more than two people then and it was private in the sense that certainly in the early days you had to be a member to get in that's the way they got around a lot of the laws was making the membership clubs which I think all to the position slightly but I couldn't tell you in exact what detail but I'm sure if the police had wanted they would have raided it book shops they used to raid regularly in those days trying to find obscene literature like pictures of men with erections that sort of thing and that was it really what attracted you to even the concept of SMGs oh well it's fact that I knew I liked pain and pleasure scenario in life and it was a good place to find like might as the court said later like minded people they made it sound horrible but in fact I thought it was wonderful so I found other people that liked the sort of SM that I enjoyed I mean I was never into beatings or anything particularly I like CBT TT and all this sort of thing was much more me bondage, well it's too much hard work but now it's doing them up and then they can't do anything I'd rather have them like loose and naked on the floor and then get on with doing what I want with them or telling them what to do this sort of thing so that was much better that's much more me now how did you qualify for membership to SMGs in those days almost everything that you had to be a member you literally signed a small bit of paper as you went in the door so it was rubbish it wasn't being a membership it was actually a way of getting round the law to say the place is private it's a bit like porno cinemas when they first started they had to say when they became all common member of the cinema or something but it was literally signed a bit of paper saying I'm a member I think there was a rule about you to wait an hour or something to make sure you're a proper member and not just coming in off the streets I mean obviously if you're going to do something you I can see that you would need to have been interested for at least an hour it's all rubbish so it was just a way of bypassing stupid rules you told me when we were preparing for this interview that you're more SM than you are leather would you please explain that how do you see the difference between these two things I think leather people enjoy not just the look but the feel and the whole experience of the dressing up and if you like the sort of power look a lot of leather guys like I mean I can give you an example of how the power looked right in the shit out of me when I was younger we'd gone off with two other friends to California for a holiday for a couple of weeks and we were in San Francisco and we decided we'd visit a bar that was just up away from where we were staying and we went into this bar and it had a sort of island serving area with high stalls around it and it was virtually filled up with head to toe leather guys with their hats and chains it actually looked brilliant you know but it scared the shit out of us because we thought this is a dangerous looking lot but there were a couple of suits still spare so we sort of tended to be like could we sit here oh yes love have a seat and after that we didn't feel at all scared so I think for a lot of the leather guys they like the look and the feel and I'm not averse to it because I used to have leather gear at one time I used to dress up but to me it wasn't the most important thing the most important thing for me was alright being in leather gear and looking like a leather guy introduced me to more people that might be into SM because a lot of SMers there's more SMers in a ratio I'd have thought in the leather scene world than there are in the ordinary game world so it was a good way of meeting other people into SM but personally once we'd done the sort of early club type interactions and things I'd rather just get all the kit off you know so it was it looked good, it felt good I could see why people really liked it I never wanted to march around as a Nazi or anything like that I don't want to people did I never quite saw that myself because given that gay folk were persecuted very badly by the SS and the Nazis I couldn't quite see the attraction of it unless you're a royalty of course in which case it's family history isn't it some of their princes have had silly days but no I still quite like leather I like the smell of leather and the feel of leather, new leather I think is a wonderful smell and that's also evocative for people people may have got into the scene in the first place may well have had some leather based experiences in childhood like either perhaps even been beaten or something with a brand new strap or you know that sort of thing where they've I mean I even used to like the smell I mean it's called satchel when it was new in fact I remember saying to someone they asked me about smoking do you smoke this is a health person do you smoke no I said I tried it when I was about 12 and I didn't like it so I gave it up I said everything I liked at 12 I'm still doing and of course smell of things I still like the same way so yes I'm sure for the leather scene it's hard of course I like motorbikes a lot of them and that was a bit of a quandary for me I wasn't the first to motorbikes per se but actually I was more of a mod in the younger days for dressing I dressed more as a mod and made a scooter and I nearly got doffed over once when I was out and about some rockers turned up on their motorbikes and were not at all friendly but I said to them and it was true I said I don't like mod music I said by and large I can't stand it I like rock and roll then the 50s and 60s rock and roll that thing and it still is so that saved me for that so that's where again that's I think the motorbike scene for a lot of leather folk come from and of course a lot of those are straight anyway so the leather scene encompasses all starters of society urges and I'm sure the look is a lot to do with it the only weakness for a particular uniform I've ever had is for the Canadian Mounties I also thought they looked wonderful and they're sort of red tunics and silly job put things in one of my hobbies I've got some Canadian friends and I wanted a higher Mounties uniform to greet them when they're cold but all you could get was fancy dress ones and they were rubbish they looked rubbish so I'm not bothering with that I wouldn't have been able to get a horse but yes I'm sure for the leather world it's the look, the feel and I'm quite happy with that I enjoyed that and I did enjoy my leather gear when I had it although I was a bit lazy with doing boots up that's another voyage having a slave of course they can do all that rubbish but you have to put mine initially and the dogs rubbish at tying up my bootlaces so I prefered I liked the actual physical sex side more than the dressing up there are two different elements there I find oh I thought you were going to expound briefly no I'm just acknowledging so one led me to more people that would enjoy the sex F7 side that I liked more they've had protest marches and things after that stuff and they had the pony cart folk they used to dress up and use them as ponies and all the little carts by the mistress that's great I've been to one or two straight SME type things and I used to go to swingers parties and things as well oh well I can feel the noose tightening even as we speak I hope I've answered that one you have how were you introduced to SME you mentioned the smells when you were younger but is there something to which you can actually pinpoint this was the beginning of it for me no not really I think the main thing goes back to where we started with a sense of loneliness I didn't have really many friends at all so if you can't play with other people you tend to play with yourself the more I did that the more I experimented and then I found that squeezing things felt good and pulling things felt good and twisting things felt good and playing with your nipples felt good so that's really where it started I can't say there's a moment when I thought oh I'm into SM and pain but due to a process of exploration as I was growing up pretty much isolated apart from during school days and things I did have a couple of clubs I started at school in order to qualify for membership you had to pop into the box and show she bum whole and it had the bonus of the sooner you applied the more you saw the more you were entitled to see all the other people that joined but they couldn't see you I got up to about four members I think and some prefects found us and turfed us out but that's the sort of thing I suppose in today's terms I'd be considered a sex mania at risk or something to anybody but I always thought it was just fun and harmless I'm pretty much sure the straight kids boys had more fun than I did in fact because they didn't worry people find it out if you see what I mean so yes it started very early days but I liked all the squeeze pull push bang squash everything else aspects of it and the more people I met that were into that the more I liked it and of course then you led into other things there was one epiphany moment that wasn't SM as such that was to do with sort of anal there was an 8mm movie some friends had once and the bloke was putting chrome balls up his bum and I thought wow that's exciting because I've never seen anything like that so you know straight back into the kitchen and see what's around and I found things around quite soon after that and that's where all the bathroom experimentation and everything started off we're not thinking back at it now it could have looked really bizarre so parents or someone had caught me but once I'd left home it didn't really matter because my mate used to go off to London to work so he'd leave quite early for a train I didn't have to leave till later so an awful lot went on before going off to work so for the benefit of the audience whoever me someday watch this video we're here primarily to explore the circumstances surrounding Operation Spanner and your involvement within that so before we get into any of the specifics around that would you give us a general idea about Operation Spanner


Imagine a whole town evacuated due to an uncontrolled underground fire. Well, imagine no more as in Pennsylvania, USA, such a place actually exists. Today we'll talk about Centralia, or more to the point the 1962 Centralia Mine Fire and its effects on the community who lived above it. Thank you to my patrons for voting for this one. It is a slight bit different from a usual plain and difficult subject, however it is still a disaster in its own right. I'm going to rate this video here on the Painted Plainly Difficult Disaster Scale. Here's a brief history of the town of Centralia for some context. Centralia is located in Columbia County in the US state of Pennsylvania, which is around here on the map. Our story starts off with the princely sum of £500, which was the combined cost of settlers buying the land that would make up Centralia from Native Americans in 1749. Unknown to the purchasers at the time, but the land would prove to have great value as it sat atop a large coal deposit. In the late 1700s, the coal beneath was discovered, however it wouldn't be for another 50 to 60 years that the deposits would be exploited. In 1832, Jonathan Faust, credited as the town's founder, opened the Bull's Head Tavern, naming the new settlement Bull's Head. Talk about effective branding. In 1842, Centralia's land was bought by the Locust Coal and Iron Company and sent Alexandra Ray, a mining engineer, to design and build a small town to accommodate the miners. Ray named the town Centerville, however the post office refused this name and instead Centralia was penned to the small town. Business started to boom after the mine run railroad and first town mines opened in 1854 and 1856 respectively. As a strange side note, Alexandra Ray was murdered in 1868 during a period of robbery and homicide that engulfed the region. A record high population of 2,761 was recorded in 1890. This would be the peak of the settlement's inhabitants. After several rights purchases and two world wars, the population had sunk to 1,986 people by 1950. An interesting and detrimental bootleg industry had developed since the financial crash of 1929 and this was the act of pillar robbing. You see, a pillar is a column of un-mined ore used to support the roof of mine or is dug out in a grid formation leaving pillars kind of how a city street system is formed. This method is called room and pillar mining. Needless to say, bootleg mining is really dangerous as it compromises destruct integrity of the mine and can and does cause collapses which make navigation of the mine system difficult. Remember this for later on. This leads us on to 1962 and the town of Centralia had an illegal dumping problem. This was due to the mainland field site being closed to the public. A disused surface strip mine that had been cleared in the 1930s had been used as one such illegal dump and the council set about clearing it up. Although it's not known for certain, most theories point to the start of the fire being linked to the clearing works of the strip mine. The local council hired a number of firemen to clear up the dump and this involved burning the rubbish even though this was against state legislation. This material was covering up a hole 15 feet wide and several feet high in the base of the north wall of the pit. The fire was started on the 27th of May 1962 and was doused with fire hoses after the material burned away quelling the visible flames. Two days after the initial rubbish burning, the flames returned. Once again, the hoses were used to douse the smouldering. A week later, fire again was spotted and a bulldozer was called up to remove some layers of waste to try and find the source of the burning. This was when the hole leading to the maze of underground mines at Centralia was discovered and it was believed that this was the fuse that lit the disaster that was the Centralia minefire. On the 2nd of July, complaints began to come into the council of foul odours coming from the area of the dump. Clarence Cashner, the head of the Independent Mines Breakermen and Truckers Union, was called in to offer some advice on how to deal with the fire problem. Cashner suggested contacting Gordon Smith, an engineer of the Department of Mines and Mineral Industries to see what could be done. Upon evaluating the strip mines, Smith suggested excavating as much of the burning rubbish as possible to prevent any more ignition of the coal seam. The price of $175 had to be authorised, causing delays to the time critical situation, thus kicking the cost into the proverbial long grass. Little did the council know, but the fire spread into the coal seams inside the abandoned mines, creating an uncontrollable inferno beneath the town streets. This was exasperated by collapsed mines from illegal pillar mining, leading to difficulties tracking the route of the fire. Tests using gas detection equipment concluded that the large hole in the pit wall and from the cracks in the north wall contained carbon monoxide concentrations, hinting at a disastrous coalmine fire. The Lee Valley Coal Company was notified of the fire, although its true cause was withheld by the council, instead stating the source to be of unknown origin during a period of unusually hot weather. A meeting on the 6th August was set up at the strip mine with the LVCC and Seschke Hannah Coal Company to estimate the cost of digging out the fire. By now, the financial burden was estimated to be near $30,000, where $175 back in July sounded like a bargain. To add some toxic cherries to the dumpster fire cake, on the 9th August deadly levels of carbon monoxide were detected closing all centralia area mines. The contract to clear the fire was put out to tender and on the 17th was awarded to Brady Inc. The company would end up excavating 53,000 cubic yards, expending just over $27,000. Brady Inc would cease excavation works on the 29th October after discovering the fire had spread beyond the area. This would necessitate more serious works such as drilling and flushing, raising the cost significantly. A new project to stop the fire was set out and would involve flushing by mixing water and crushed rock into holes drilled ahead of the predicted path of the fire, and this was to fill voids and reduce oxygen. The new project was estimated to cost $40,000 and on the 19th November, K and H excavating began drilling some 84 holes. This would last until March 1963 when the money ran out after $43,000 was spent. Not to talk surprisingly, the fire was not contained. By May 1963 the fire had spread approximately 700 feet eastward and the fire had heated up as seen by steam plumes emanating from the ground. Three plans were drawn up and submitted to the Secretary of Mines and Mineral Industries. Option 1 costing $296,000 consisted of entrenching the fire and backfilling the trench with incombustible material such as clay. Option 2 costing $151,000 offered a smaller trench in an incomplete circle followed by extensive flushing. Option 3 was to drill more bore holes for flushing and would cost around $82,000. However, without proper funding the projects were all abandoned, instead opting to dig a cheaper trench. However, by October the fire was discovered on either side of the incomplete attempt, costing in excess of $32,000. In total the Council has spent over $100,000 with no effective control of fire, essentially dooming the fate of Centralia. Work would not restart until November 1966 when federal state funding was secured and after a year spending another $326,000 it was again discovered that the fire had spread. Seven more projects would be undertaken in 1970, 1973 and 1980, including flushing, injecting fly ash and the digging of containment trenches, all of which would not effectively control the fire. The ground beneath the town itself increased in heat, reaching over 900 degrees Fahrenheit. Smoke poured from sinkholes and filled basements with gas necessitating evacuation of residents due to oxygen deficiencies and heightened levels of carbon monoxide. Increased reports of health issues plagued the residents with heightened respiratory complications. Locals became aware of the severity of the fire when the roads began to bulge and sinkholes appeared. Residents were divided between staying and abandoning the town. In 1982 the US Congress allocated $42 million for the relocation of Centralia's inhabitants. Unsurprisingly many took the money and ran, as essentially the property was now worthless. In total 1,000 people moved out of the town and 500 buildings were raised at the ground, leaving behind around 63 residents. Many buildings foundations became unstable, as the fire burned away creating voids beneath the town, meaning demolition was the only route left. Pennsylvania Governor Bob Casey invoked eminent domain on the town in 1992, condemning any remaining structures. The town was now all but gone except for a handful of steadfast residents. The order refused any rights, the order refused the rights of any resident to hand down or sell their property. By 2003 the US Postal Service revoked the town zip code and after 2009 evictions cleared out most of who remained. However Centralia today still exists with a population of 5. After 20 years of battling over eminent domain in 2013, the remaining residents were allowed to stay and part of the settlement gave a grand sum of $349,500 to the 8 litigants. The town still receives a large amount of visitors in a strange form of abandoned expolation, and the main attraction of sorts being the Graffiti Highway which was once part of PA Route 61. Havis now has been covered up with dirt to deter visitors during the Covid pandemic. The fire is predicted to burn for at least another century as the seams of coal in the region act as a limitless source of fuel. I hope you enjoyed the video, if you would like to support the channel financially you can on Patreon from $1 per and that gets you early access to videos and access to votes. I have a YouTube membership as well from 99p per month and that gets you early access to videos. Check me out on Twitter and also if you'd like to wear my merch you can purchase it at my Teespring store and all that's left to say is thank you for watching.