论文标题
两类狭窄的BCH代码及其双重
Two classes of narrow-sense BCH codes and their duals
论文作者
论文摘要
BCH代码及其双重代码是循环代码的两个特殊子类，在许多情况下是最佳的线性代码。在研究BCH循环代码方面的许多进展中已经取得了很多进展，但是对于BCH代码的双重距离的最小距离知之甚少。最近，引入了一个称为双BCH代码的新概念，以调查BCH代码的双重二重奏和在\ cite {gdl21}中的最小距离上的下限。对于Prime Power $ Q $和整数$ M \ GE 4 $，让$ n = \ frac {q^m-1} {q+1} $ \（$ m $ even）或$ n = \ frac {q^m-1} {q-1} {q-1} {q-1} $ \（$ q> 2 $）。 在本文中，将提供一些足够和必要的条件，以确保长度$ n $的窄态BCH代码为双BCH代码，这将结果扩展为\ cite {gdl21}。以$ n = \ frac {q^m-1} {q+1} $ \（$ m $偶数）开发了双重代码的最小距离的下限。作为副产品，我们介绍了两种类型的最大coset领导者$δ_1$ modulo $ n $，这证明了\ cite {wlp19}中的猜想，并部分解决了\ cite {li2017}的开放问题。我们还研究了带有设计距离$Δ_1$的长度$ n $的窄态BCH代码的参数。本文介绍的BCH代码通常具有良好的参数。
BCH codes and their dual codes are two special subclasses of cyclic codes and are the best linear codes in many cases. A lot of progress on the study of BCH cyclic codes has been made, but little is known about the minimum distances of the duals of BCH codes. Recently, a new concept called dually-BCH code was introduced to investigate the duals of BCH codes and the lower bounds on their minimum distances in \cite{GDL21}. For a prime power $q$ and an integer $m \ge 4$, let $n=\frac{q^m-1}{q+1}$ \ ($m$ even), or $n=\frac{q^m-1}{q-1}$ \ ($q>2$). In this paper, some sufficient and necessary conditions in terms of the designed distance will be given to ensure that the narrow-sense BCH codes of length $n$ are dually-BCH codes, which extended the results in \cite{GDL21}. Lower bounds on the minimum distances of their dual codes are developed for $n=\frac{q^m-1}{q+1}$ \ ($m$ even). As byproducts, we present the largest coset leader $δ_1$ modulo $n$ being of two types, which proves a conjecture in \cite{WLP19} and partially solves an open problem in \cite{Li2017}. We also investigate the parameters of the narrow-sense BCH codes of length $n$ with design distance $δ_1$. The BCH codes presented in this paper have good parameters in general.
