WEIGHT POLYNOMIALS OF SELF-DUAL CODES AND THE MacWILLIAMS IDENTITIES

Many error correcting codes are known to be self-dual. Hence the MacWilliams identities put a considerable restriction on the possible weight distribution of such a code. We show that this restriction, for codes over GF(2) and GF(3), is that the weight polynomial must lie in an explicitly described free polynomial ring. To extend these results (in part) to self-dual codes over larger fields, we introduce more general weight polynomials and extend the MacWilliams identities to these. 1. Let F be a finite field with q elements and let F be the direct product of F with itself n times regarded as a vector space as usual. A /^-dimensional linear subspace A of F is sometimes called an (n , fc)-code. If v = < v, , . . . , vn > e F , then the weight of v is the number of indices i for which v{ ¥= 0. Suppose A is an (n , fc)-code. For each i = 0, . . . , n let wt be the number of vectors in A having weight i. Then the homogeneous polynomial WA £ C [S, T] given by WA = S w , 5 n T ' is called the weight polynomial of A. Let ( , ) denote the usual inner product F x F -> F. IÎA is a linear subspace of F, its dual is B = {vEF : ( V x E ^ ) ( v , x ) = 0}. The MacWilliams identities [2] connect the weight polynomial WA of A with the weight polynomial of WB of the dual B. They can be expressed in the single equation (1) qWA =WB(S + (q-\)T9S-T)9 where the right hand side means the result of replacing S and T in WB by S + (q — 1) T and S — T respectively. It may happen that A is identical with its dual B ; that is, A may be self-dual. This clearly implies that n = 2k. If n is even and q ^ — 1 (mod 4) then F always contains self-dual subspaces. If q = — 1 (mod 4), then self-dual subspaces appear if and only if n is divisible by 4.