By N.J.A. Sloane

ISBN-10: 3211813039

ISBN-13: 9783211813034

ISBN-10: 3709128641

ISBN-13: 9783709128640

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11) shows that the codewords ofweight 8 in the extended Golay codeform a 5-(24,8,1) design. 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.

MacWilliams, says that the weigth cnumeration of the dual code ct1 is completely determined just by the weight cnumerator of CC. M. Gleason, states that the weight enumerator of any sclf-dual code (in which the weigth of any codeward isamultiple of 4) is a polynomial in the weigth enumerators of the extended Hamming code and the extended Golay code. This is an extremely powerful theorem for finding the minimum distance of large self-dual codes. lt can also be used to show that certain codes da not exist.

6 says that a polynomial basis for &'(G) can always be found. ) is any polynomial then h<~> = is an invariant of G. Proof. For any A' € G. ) AEG ( 4o 4 8) 0 AEG = .!. g L f(Ax) AEG = h(x) since the last sum is a rearrangement of the one before. D. Furthermore, it is clear that all invariants of G can be obtained in this way. 6 shows that a polynomial basis for the invariants of G can be obtained by averaging over G all monomials. bl b2 x1 x2 bn 000 ~ of total degree l:bi ~ g o More generally, any symmetric function of the g polynomials {f(Ax);AE G} is an invariant of G.

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

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