发表机构
Kenyon College; IIT Gandhinagar（肯扬学院; 甘地纳加尔印度理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文解决了 $\mathbb{Z}_4$ 上奇数长循环码构造准循环码时类型继承的开放问题，给出充要条件，并发现多个 Lee 距离更优的新码。
AI 中文摘要
给定 $\mathbb{Z}_4$ 上奇数长度 $m$ 的循环码 $C_g = \langle g(x) \rangle$，构造准循环（QC）码的一种常见方法是选取 $f_1, \dots, f_\ell \in \mathbb{Z}_4[x]$ 并令 $C = \langle (f_1 g, \dots, f_\ell g) \rangle$。由于 $\mathbb{Z}_4$ 不是域，$C_g$ 的类型（$4^{k_1}2^{k_2}$）不一定被 $C$ 继承。确定 $C_g$ 的类型被 $C$ 继承的条件在最近的文章 [AydinLuOnta2023] 中被作为一个开放问题提出。本文解决了该问题。我们首先给出类型保持的两个充分条件：一个要求某个 $f_i$ 在 $\mathbb{F}_2$ 上与 $x^m-1$ 互素，另一个仅要求 $f_i$ 整体上与 $x^m-1$ 互素。这两个条件在一般情况下都不是必要的。利用奇数 $m$ 时 $x^m-1$ 无平方因子的性质，我们通过中国剩余定理（CRT）将 $\mathbb{Z}_4[x]/\langle x^m-1\rangle$ 分解为有限链环的乘积，并推导出 $f_1, \dots, f_\ell$ 的一个条件，该条件是 $C$ 匹配 $C_g$ 类型的充分必要条件。该条件仅依赖于 $x^m-1$ 中 $g$ 未消失的不可约因子。这也为 $C$ 是自由 $\mathbb{Z}_4$-模提供了简单检验。最后，我们报告了由 Magma 软件 [Magma1997] 依据该准则进行计算机搜索发现的许多新的 $\mathbb{Z}_4$ 上 QC 码，其 Lee 距离大于先前已知的同类型码。
英文摘要
Given a cyclic code $C_g = \langle g(x) \rangle$ of odd length $m$ over $\mathbb{Z}_4$, one common way to build a quasi-cyclic (QC) code is to pick $f_1, \dots, f_\ell \in \mathbb{Z}_4[x]$ and let $C = \langle (f_1 g, \dots, f_\ell g) \rangle$. Because $\mathbb{Z}_4$ is not a field, the type of $C_g$ ($4^{k_1}2^{k_2}$) is not necessarily inherited by $C$. Determining conditions under which the type of $C_g$ is inherited by $C$ was posed as an open problem recently in \cite{AydinLuOnta2023}. In this paper, we settle this problem. We first give two sufficient conditions for the type to be preserved: one requires a single $f_i$ to be coprime to $x^m-1$ over $\mathbb{F}_2$, the other only requires the $f_i$ to be jointly coprime to it. Neither condition is necessary in general. Using the fact that $x^m-1$ is squarefree for odd $m$, we decompose $\mathbb{Z}_4[x]/\langle x^m-1\rangle$ into a product of finite chain rings using Chinese remainder theorem (CRT) and derive a condition on $f_1, \dots, f_\ell$ that is both necessary and sufficient for $C$ to match the type of $C_g$. This condition depends only on the irreducible factors of $x^m-1$ where $g$ does not already vanish. This also yields a simple test for when $C$ is a free $\mathbb{Z}_4$-module. Finally, we report many new QC codes over $\mathbb{Z}_4$, found by computer searches using Magma software~\cite{Magma1997} guided by this criterion, with Lee distances greater than previously known codes of the same type.
