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

Authors:
T. Shaska and G. S. Wijesiri

Journal:
Proc. Amer. Math. Soc. **136** (2008), 849-857

MSC (2000):
Primary 11H71, 94B75; Secondary 11H31

DOI:
https://doi.org/10.1090/S0002-9939-07-09152-6

Published electronically:
December 3, 2007

MathSciNet review:
2361856

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Abstract | References | Similar Articles | Additional Information

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

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Additional Information

**T. Shaska**

Affiliation:
Department of Mathematics and Statistics, Oakland University, 368 Science and Engineering Building, Rochester, Michigan 48309.

MR Author ID:
678224

ORCID:
0000-0002-2293-8230

Email:
shaska@oakland.edu

**G. S. Wijesiri**

Affiliation:
Department of Mathematics and Statistics, Oakland University, 368 Science and Engineering Building, Rochester, Michigan 48309

Email:
gwijesi@oakland.edu

Keywords:
Theta functions,
Hermitian lattices,
codes.

Received by editor(s):
January 10, 2007

Received by editor(s) in revised form:
February 14, 2007, February 21, 2007, and February 24, 2007

Published electronically:
December 3, 2007

Communicated by:
Wen-Ching Winnie Li

Article copyright:
© Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.