Derivatives of Complete Weight Enumerators and New Balance Principle of Binary Self-Dual Codes
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Let H be the standard Hadamard matrix of order two and let K=2^-1/2H. It is known that the complete weight enumerator W of a binary self-dual code of length n is an eigenvector corresponding to an eigenvalue 1 of the Kronecker power K^[n]. For every integer t in the interval [0,n] we define the derivative of order t, W_<t>, of W in such a way that W_<t> is in the eigenspace of 1 of the matrix K^[n-t]. For large values of t, W_<t> contains less information about the code but has smaller length while W_<0>=W completely determines the code. We compute the derivative of order n-5 for the extended Golay code of length 24, the extended quadratic residue code of length 48, and the putative [72,24,12] code and show that they are in the eigenspace of 1 of the matrix prove a new balance equation which involves the number of code vectors of given weight having 1 in a selected coordinate position. As an example, we use the balance equation to eliminate some candidates for weight enumerators of binary self-dual codes of length eight.
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