 Engineers, who use the codes for data transmission, and coding theorists, who analyse extremal properties of codes, typically want the rate of the code to be as high as possible, even if this means that the construction becomes mathematically slightly less elegant.On the other hand, for many applications of Hadamard codes in theoretical computer science it is not so important to achieve the optimal rate, and hence simpler constructions of Hadamard codes are preferred since they can be analyzed more elegantly.When given a binary message formula_14 of length formula_8, the Hadamard code encodes the message into a codeword formula_16 using an encoding function formula_17
This function makes use of the inner product formula_18 of two vectors formula_19, which is defined as follows