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arXiv 1404.7771math.CO
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关于次闭类中渐近好线性码的存在性

On the existence of asymptotically good linear codes in minor-closed classes

Peter Nelson, Stefan H. M. van Zwam

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摘要：

设$\mathcal{C} = (C_1, C_2, \ldots)$是一个码序列，使得每个$C_i$是某个固定有限域$\mathbb{F}$上的线性$[n_i,k_i,d_i]$码，其中$n_i$是码字的长度，$k_i$是维数，$d_i$是最小距离。如果对于某些$\varepsilon > 0$和所有$i$、$n_i \geq i$、$k_i/n_i \geq \varepsilon$和$d_i/n_i \geq \varepsilon$，我们说$\mathcal{C}$是渐近好的。渐近好码序列是存在的。证明了若$\mathcal{C}$是一类GF $(p^n)$ -线性码（其中$p$是素数，$n \geq 1$是素数），在删截和缩短下是闭的，且若$\mathcal{C}$包含渐近好序列，则$\mathcal{C}$必包含所有GF $(p)$ -线性码.我们的证明依赖于一个强大的新的结果拟阵结构理论。

英文摘要：

Let $\mathcal{C} = (C_1, C_2, \ldots)$ be a sequence of codes such that each $C_i$ is a linear $[n_i,k_i,d_i]$-code over some fixed finite field $\mathbb{F}$, where $n_i$ is the length of the codewords, $k_i$ is the dimension, and $d_i$ is the minimum distance. We say that $\mathcal{C}$ is asymptotically good if, for some $\varepsilon > 0$ and for all $i$, $n_i \geq i$, $k_i/n_i \geq \varepsilon$, and $d_i/n_i \geq \varepsilon$. Sequences of asymptotically good codes exist. We prove that if $\mathcal{C}$ is a class of GF$(p^n)$-linear codes (where $p$ is prime and $n \geq 1$), closed under puncturing and shortening, and if $\mathcal{C}$ contains an asymptotically good sequence, then $\mathcal{C}$ must contain all GF$(p)$-linear codes. Our proof relies on a powerful new result from matroid structure theory.

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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