On the parameters of some LCD BCH codes over $$\mathbb {F}_q$$ with length $$(q^m+1)/\lambda $$

Cryptography and Communications Pub Date : 2024-01-26 DOI:10.1007/s12095-024-00697-z

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Abstract

As a particular subclass of cyclic codes, BCH codes have wide applications in storage devices, communication systems, consumer electronics and other fields. However, parameters of BCH codes are unknown in general. In this paper, we investigate parameters of BCH codes of length $\frac{q^m+1}{\lambda }$ where $\lambda \mid q+1$ .Some new techniques are employed to study the coset leaders. For any odd prime power q and $m=4,8$ , or $m\ge 12$ and $m\equiv 4~ (\textrm{mod}~ 8)$ , the second, the third and the fourth largest coset leaders modulo $q^m+1$ are determined, and the dimensions of some BCH codes of length $q^m+1$ with large designed distances are given. For $1<\lambda <q+1$ , the first few largest coset leaders and the coset leaders modulo $\frac{q^m+1}{\lambda }$ in the range 1 to $ \frac{ q^{\lfloor (m+1)/2\rfloor }}{\lambda }$ are studied, and the dimensions of some BCH codes of length $\frac{q^m+1}{\lambda }$ are given as well. The BCH codes presented in this paper are LCD codes and have a sharper lower bound on the minimum distance than the well-known BCH bound.

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关于长度为 $$(q^m+1)/\lambda $$ 的 $$\mathbb {F}_q$$ 上一些 LCD BCH 编码的参数

摘要 BCH 码作为循环码的一个特殊子类，在存储设备、通信系统、消费电子产品等领域有着广泛的应用。然而，一般情况下 BCH 码的参数是未知的。本文研究了长度为 $\frac{q^m+1}{lambda }$ 的 BCH 码的参数，其中 $\lambda \mid q+1$ .采用了一些新技术来研究余集领导者。对于任何奇素数幂q和（m=4,8），或者（m≥12）和（m≥4~ (textrm{mod}~8)），第二、第三和第四幂都是奇数幂。确定了模（q^m+1）的第二、第三和第四大余弦组长，并给出了一些具有大设计距离的长度为（q^m+1）的 BCH 码的尺寸。对于 $1<\lambda <；q+1) 时，研究了在 1 到 \( \frac{ q^{\lfloor (m+1)/2\rfloor }}{\lambda }$ 范围内的前几个最大的子集领导者和 modulo $\frac{q^m+1}{\lambda }$ 的子集领导者，并给出了一些长度为 $\frac{q^m+1}{\lambda }$ 的 BCH 码的维数。本文提出的 BCH 编码是 LCD 编码，其最小距离的下界比著名的 BCH 界值更小。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Cryptography and Communications

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