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Codes over F_{p²} and F_p×F_p, lattices, and theta functions

Tanush Shaska, Caleb Shor

I-AIMS paper 2006-3 · Published in Ser. Coding Theory Cryptol. 3, World Sci. (2007), 70–80 · 2007

Let ℓ > 0 be a square free integer and 𝒪_K the ring of integers of the imaginary quadratic field K = ℚ(√(−ℓ)). Codes C over K determine lattices Λ_ℓ(C) over rings 𝒪_K/p𝒪_K. The theta functions θ_(Λ_ℓ(C)) of such lattices are known to determine the symmetrized weight enumerator swe(C) for small primes p = 2, 3. In this paper we explore such constructions for any p. If p ∤ ℓ then the ring R:= 𝒪_K/p𝒪_K is isomorphic to 𝔽_(p²) or 𝔽ₚ × 𝔽ₚ. Given a code C over R we define new theta functions on the corresponding lattices. We prove that the theta series θ_(Λ_ℓ(C)) can be written in terms of the complete weight enumerator of C and that θ_(Λ_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 θ_(Λ_ℓ(C)).

MSC classes: Primary 94B27; Secondary 11F27, 94B05, 11T71

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