Derivatives of Complete Weight Enumerators and New Balance Principle of Binary Self-Dual Codes

Let H be the standard Hadamard matrix of order two and let K=2^{-1/2}H. 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 $% K^{[5]}.$ We use the derivatives to 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.