Two-Tuples Weight Enumerators

Abstract

We consider an inner product of a special type in the space of $n$-tuples over a finite field $F_q$, of characteristic $p$. We prove that there is a very close relationship between the self-dual $q$-ary additive codes under this inner product and the self-dual $p$-ary codes under the usual dot product. We prove the MacWilliams identities for complete weight enumerators of $q$-ary additive codes with respect to the new inner product. We define a two-tuple weight enumerator of a binary self-dual code and prove that it is invariant of a group of order 384. We compute the Molien series of this group and find a good polynomial basis for the ring of its invariants.