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arXiv 1209.0469math.AG
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尺寸为4的环、厄米特格和相应的θ函数上的码

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

T. Shaska, G. S. Wijesiri

发表信息：
Proc. Amer. Math. Soc. 136 (2008), no. 3, 849--857
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摘要：

设$K=Q(\sqrt{-\ell})$是一个具有整数环$Ø_K$的虚二次域，其中$\ell$是一个无平方的整数，使得$\ell\equiv 3 \mod 4$和$C=[n, k]$是定义在$Ø_K/2Ø_K$上的线性码. $C$的能级$\ell$ θ函数$\Th_{Ł_{\ell} (C)} $定义在格点$Ł_{\ell} (C):= \set {x \in Ø_K^n : ρ_\ell (x) \in C}$上，其中$ρ_{\ell}:Ø_K \rightarrow Ø_K/2Ø_K$是自然投影。在本文中，我们证明了：i）对于任何$\ell, \ell^\prime$，使得$\ell \leq \ell^\prime$，$\Th_{Λ_\ell}(q)$和$\Th_{Λ_{\ell^\prime}}(q)$具有相同的系数直到$q^{\frac {\ell+1}{4}}$，ii）对于$\ell \geq \frac {2(n+1)(n+2)}{n} -1$，$\Th_{Ł_{\ell}} (C)$唯一地确定代码$C$，iii）对于$\ell < \frac {2(n+1)(n+2)}{n} -1$，存在正维族的对称权枚举多项式对应于$\Th_{\La_\ell}(C)$ .

英文摘要：

Let $K=Q(\sqrt{-\ell})$ be an imaginary quadratic field with ring of integers $Ø_K$, where $\ell$ is a square free integer such that $\ell\equiv 3 \mod 4$ and $C=[n, k]$ be a linear code defined over $Ø_K/2Ø_K$. The level $\ell$ theta function $\Th_{Ł_{\ell} (C)} $ of $C$ is defined on the lattice $Ł_{\ell} (C):= \set {x \in Ø_K^n : ρ_\ell (x) \in C}$, where $ρ_{\ell}:Ø_K \rightarrow Ø_K/2Ø_K$ is the natural projection. In this paper, we prove that: % i) for any $\ell, \ell^\prime$ such that $\ell \leq \ell^\prime$, $\Th_{Λ_\ell}(q)$ and $\Th_{Λ_{\ell^\prime}}(q)$ have the same coefficients up to $q^{\frac {\ell+1}{4}}$, % ii) for $\ell \geq \frac {2(n+1)(n+2)}{n} -1$, $\Th_{Ł_{\ell}} (C)$ 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 $\Th_{\La_\ell}(C)$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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