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Proceedings of the American Mathematical Society
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
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.
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

DOI: http://dx.doi.org/10.1090/S0002-9939-07-09152-6
PII: S 0002-9939(07)09152-6
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.