Narrow Results

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).

Historically ideal theory for lattices was developed by Hashimoto. He established that there is a one to one correspondence between ideals and congruence relations of a lattice L under which the ideal corresponding to a congruence relation is a whole congruence class under it if and only if L is a generalized Boolean algebra. His proof involved topological ideas which were later simplified, using lattice theoretic ideas, by Gratzer and Schmidt. Also Gratzer introduced the notion of standard elements and ideals in lattices which were extensively studied by Gratzer and Schmidt. It was shown that standard ideals of lattices play a role somewhat similar to that of normal subgroups of groups or ideals of rings. Later, Fried and Schmidt extended the notion of standard ideals of lattices to convex sublattices. Generalizations of some of these results to trellises (or also called weakly associative lattices) may be found in Bhatta and Ramananda and Shashirekha.