[论文解读] Characterisation of a family of neighbour transitive codes

Neil I. Gillespie, Cheryl E. Praeger|arXiv (Cornell University)|May 21, 2014

Coding theory and cryptography参考文献 42被引用 9

一句话总结

该论文表征了一类最小距离至少为3、字母表几乎单态且与哈密顿图自同构群基群非平凡相交的邻域传递码。证明此类码要么等价于某个特定子码，要么是其$X$-平移的不相交并集，且该子码为邻域传递的频率置换阵列。

ABSTRACT

We consider codes of length $m$ over an alphabet of size $q$ as subsets of the vertex set of the Hamming graph $Γ=H(m,q)$. A code for which there exists an automorphism group $X\leq Aut(Γ)$ that acts transitively on the code and on its set of neighbours is said to be neighbour transitive, and were introduced by the authors as a group theoretic analogue to the assumption that single errors are equally likely over a noisy channel. Examples of neighbour transitive codes include the Hamming codes, various Golay codes, certain Hadamard codes, the Nordstrom Robinson codes, certain permutation codes and frequency permutation arrays, which have connections with powerline communication, and also completely transitive codes, a subfamily of completely regular codes, which themselves have attracted a lot of interest. It is known that for any neighbour transitive code with minimum distance at least 3 there exists a subgroup of $X$ that has a $2$-transitive action on the alphabet over which the code is defined. Therefore, by Burnside's theorem, this action is of almost simple or affine type. If the action is of almost simple type, we say the code is alphabet almost simple neighbour transitive. In this paper we characterise a family of neighbour transitive codes, in particular, the alphabet almost simple neighbour transitive codes with minimum distance at least $3$, and for which the group $X$ has a non-trivial intersection with the base group of $Aut(Γ)$. If $C$ is such a code, we show that, up to equivalence, there exists a subcode $Δ$ that can be completely described, and that either $C=Δ$, or $Δ$ is a neighbour transitive frequency permutation array and $C$ is the disjoint union of $X$-translates of $Δ$. We also prove that any finite group can be identified in a natural way with a neighbour transitive code.

研究动机与目标

• 表征最小距离至少为3、且与哈密顿图自同构群基群有非平凡交的字母表几乎单态邻域传递码。
• 通过分析交集群$K = X \bigcap B$的根基，确定此类码的结构。
• 证明在等价意义下，码要么等于已知的$\text{soc}(K)$-轨道，要么是其$X$-平移的不相交并集。
• 通过构造性嵌入建立有限群与邻域传递码之间的联系。
• 利用群论与组合工具，为一大类邻域传递码提供分类框架。

提出的方法

• 利用自同构群$X \trianglelefteq \text{Aut}(\text{哈密顿图})$，通过在码字及其邻点上的传递作用，定义$X$-邻域传递码。
• 利用伯恩赛德定理将字母表上的$2$-传递作用分类为仿射型或几乎单型，重点研究后者。
• 应用斯科特引理分析$\text{soc}(K)$的结构，其中$K = X \bigcap B$，证明其为非交换单群的次直积。
• 构造并分析码$C$的投影码，以确定在$X$-不变分划结构下的可能配置。
• 使用乘积与重复构造方法生成实例，并验证最小距离与邻域传递性等性质。
• 对$\text{soc}(K)$进行轨道分析，识别出一个规范子码$\triangleq \text{soc}(K)$-轨道，该轨道完全描述了码在等价意义下的结构。

实验结果

研究问题

• RQ1 字母表几乎单态$X$-邻域传递码在最小距离至少为3且与基群有非平凡交时，其结构分类是什么？
• RQ2 如何利用交集群$K = X \bigcap B$的根基来表征码的结构？
• RQ3 在何种条件下，邻域传递码是某个频率置换阵列的$X$-平移的不相交并集？
• RQ4 每个有限群是否都能自然地嵌入为哈密顿图中的邻域传递码？
• RQ5 沿$X$-不变分划投影此类码时，最小距离会发生什么变化？

主要发现

• 任何字母表几乎单态$X$-邻域传递码，若$\text{min distance} \neq 3$且$K = X \bigcap B \neq 1$，则等价于一个码，其要么等于其$\text{soc}(K)$-轨道，要么是该轨道的$X$-平移的不相交并集。
• $\text{soc}(K)$-轨道$\triangleq \text{soc}(K)$-轨道被显式描述，且为邻域传递的频率置换阵列。
• 证明码$C$是$\triangleq \text{soc}(K)$-轨道的$X$-平移的不相交并集，其中平移的指标由$A_q$在$S_q$中的陪集代表元集合$\tau$给出。
• 构造了一个实例，其中投影码的最小距离为2，而原码的最小距离为3，表明投影可能降低最小距离。
• $\text{soc}(K)$被证明同构于$\text{Prod}_\nu(C(A_q))$，即$A_q$的对角子群的积，且等于相关群$G$的$\text{soc}(G)$。
• 每个有限群均可通过基于对称群与交错群乘积的构造，自然地嵌入为邻域传递码，从而在有限群与此类码之间建立自然对应关系。