The biweight enumerator of self-orthogonal binary codes

Let C be a binary code of length n and let J_C (a, b, c, d) be its biweight enumerator. If n is even and C is self-dual, then J_C is an element of the ring R^$\text{G}$ of absolute invariants of a certain group $\text{G}$. Under the additional assumption that all codewords of C have weight divisible by 4, a similar result holds with a different group. If n is odd and C is maximal self-orthogonal, then J_C is an element of a certain R^$\text{G}$-module. Again a similar result holds if the codewords of C have weights divisible by 4. The groups involved are related to finite groups generated by reflections. In this paper the structure of these groups is described, and polynomial bases for the rings and modules in question are obtained. This answers a question posed in The Theory of Error- correcting Codes by F.J. MacWilliams and N.J.A. Sloane.