Codes over rings of size four, Hermitian lattices, and corresponding theta functions

Abstract

Let K = Q ( − ℓ ) K=Q(\sqrt {-\ell }) be an imaginary quadratic field with ring of integers O K \mathcal {O}_K , where ℓ \ell is a square free integer such that ℓ ≡ 3 mod 4 \ell \equiv 3 \mod 4 , and let C = [ n , k ] C=[n, k] is a linear code defined over O K / 2 O K \mathcal {O}_K/2\mathcal {O}_K . The level ℓ \ell theta function Θ Λ ℓ ( C ) \Theta _{\Lambda _{\ell } (C) } of C C is defined on the lattice Λ ℓ ( C ) := { x ∈ O K n : ρ ℓ ( x ) ∈ C } \Lambda _{\ell } (C):= \{ x \in \mathcal {O}_K^n : \rho _\ell (x) \in C\} , where ρ ℓ : O K → O K / 2 O K \rho _{\ell }:\mathcal {O}_K \rightarrow \mathcal {O}_K/2\mathcal {O}_K is the natural projection. In this paper, we prove that: i) for any ℓ , ℓ ′ \ell , \ell ^\prime such that ℓ ≤ ℓ ′ \ell \leq \ell ^\prime , Θ Λ ℓ ( q ) \Theta _{\Lambda _\ell }(q) and Θ Λ ℓ ′ ( q ) \Theta _{\Lambda _{\ell ^\prime }}(q) have the same coefficients up to q ℓ + 1 4 q^{\frac {\ell +1}{4}} , ii) for ℓ ≥ 2 ( n + 1 ) ( n + 2 ) n − 1 \ell \geq \frac {2(n+1)(n+2)}{n} -1 , Θ Λ ℓ ( C ) \Theta _{\Lambda _{\ell }} (C) determines the code C C uniquely, iii) for ℓ > 2 ( n + 1 ) ( n + 2 ) n − 1 \ell > \frac {2(n+1)(n+2)}{n} -1 , there is a positive dimensional family of symmetrized weight enumerator polynomials corresponding to Θ Λ ℓ ( C ) \Theta _{\Lambda _\ell }(C) .