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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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: 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.