Codes over Fp2 and Fp x Fp, lattices, and theta functions

Let l > 0 be a square free integer and O-K the ring of integers of the imaginary quadratic field K = Q (root-1). Codes C over K determine lattices A(l)(C) over rings O-K/pO(K). The theta functions theta(Lambda l),(C) of such lattices axe known to determine the symmetrized weight enumerator swe(C) for small primes p = 2,3; see [1, 10]. In this paper we explore such constructions for any p. If p inverted iota l then the ring R := O-K/pO(K) is isomorphic to F-p2 or F-p x F-p. Given a code C over R we define new theta functions on the corresponding lattices. We prove that the theta series theta(Lambda l) (C) can be written in terms of the complete weight enumerator of C and that theta(Lambda l) (C) is the same for almost all l. Furthermore, for large enough l, there is a unique complete weight enumerator polynomial which corresponds to theta(Lambda l) (C) .