Codes over F_p² and F_p × F_p, lattices, and theta functions

Let ℓ > 0 be a square free integer and the ring of integers of the imaginary quadratic field . Codes C over K determine lattices Λ_ℓ(C) over rings . The theta functions θ_Λℓ(C) of such lattices are 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 ∤ ℓ then the ring is isomorphic to 𝔽_p² or 𝔽_p × 𝔽_p. Given a code C over we define new theta functions on the corresponding lattices. We prove that the theta series θ_Λℓ(C) can be written in terms of the complete weight enumerator of C and that θ_Λℓ(C) is the same for almost all ℓ. Furthermore, for large enough ℓ, there is a unique complete weight enumerator polynomial which corresponds to θ_Λℓ(C).