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

Tanush Shaska, G. S. Wijesiri

I-AIMS paper 2006-2 · Published in Proc. Amer. Math. Soc. 136 (2008), 849–857 · 2008

Let K = ℚ(√(−ℓ)) be an imaginary quadratic field with ring of integers 𝒪_K, where ℓ is a square free integer such that ℓ ≡ 3 mod 4, and let C = [n, k] be a linear code defined over 𝒪_K/2𝒪_K. The level ℓ theta function Θ_(Λ_ℓ(C)) of C is defined on the lattice Λ_ℓ(C):= {x ∈ 𝒪_Kⁿ: ρ_ℓ(x) ∈ C}, where ρ_ℓ: 𝒪_K → 𝒪_K/2𝒪_K is the natural projection. In this paper, we prove that: i) for any ℓ, ℓ′ such that ℓ ≤ ℓ′, Θ_(Λ_ℓ)(q) and Θ_(Λ_(ℓ′))(q) have the same coefficients up to q^((ℓ+1)/4); ii) for ℓ ≥ (2(n+1)(n+2))/n − 1, Θ_(Λ_ℓ)(C) determines the code C uniquely; iii) for ℓ < (2(n+1)(n+2))/n − 1 there is a positive dimensional family of symmetrized weight enumerator polynomials corresponding to Θ_(Λ_ℓ)(C).

MSC classes: Primary 11H71, 94B75; Secondary 11H31

PDF

References

1. K. S. Chua, Codes over GF(4) and F2 ×F2 and Hermitian lattices over imaginary quadratic fields, Proc. Amer. Math. Soc. 133 (2005), no. 3, 661–670 (electronic).
2. N. J. A. Sloane, Codes over GF(4) and complex lattices, J. Algebra 52 (1978), no. 1, 168–181.
3. J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, Second, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290, Springer-Verlag, New York, 1993.
4. H. H. Chan K. S. Chua and P. Solé, Seven-modular lattices and a septic base Jacobi identity, J. Number Theory 99 (2003), no. 2, 361–372.

All preprints