Download A Short Course on Error Correcting Codes by N.J.A. Sloane PDF

By N.J.A. Sloane

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Xn-r -l g(x) are linearly independent elements of ~. The corresponding vectors are the rows of G. Thus there are 2n- r distinct vectors in ~ of the form b(x)g(x), deg b(x)

Another proof of this result follows from Assmus and Mattson's theorem [60] (which we do not have time to prove here), which says that if a code contains weights 0, T1 ,T2, ••• ,T8 ,nand the dual code has minimum weight d'>s, then the codewords of weight Ti form a (d' -s)-design, for i = 1, ... ,s. 14) Theorem The codewords of weights 8, 12, and 16 in the extended Golay codeform 5 - ( 24 '8' 1) 5 - (24,12,48) 5 - (24,16,78) designs, respectively. 15) Theorem Let u be a codeward of ~ of weight 8, with 1's in coordinates a 1,a2 , ••• ,as.

In IRn this says r(x) = b(x)g(x)€~, a contradiction unless r(x) = 0. (d), (e): Clearly g(x),xg(x), ... ,xn-r -l g(x) are linearly independent elements of ~. The corresponding vectors are the rows of G. Thus there are 2n- r distinct vectors in ~ of the form b(x)g(x), deg b(x)