Two classes of LCD BCH codes over finite fields

TL;DR

The paper studies LCD q-ary BCH codes of lengths $n=\frac{q^{m}+1}{q+1}$ and $n=q^{m}+1$, establishing LCD properties and deriving dimension formulas for narrow-sense BCH codes with design distance $\delta=\ell q^{\frac{m-1}{2}}+1$. It identifies coset leaders and provides explicit dimension results, including special cases for binary length $n=\frac{2^{m}+1}{3}$ and ternary antiprimitive codes, where it yields a lower bound on the dual distance and demonstrates several optimal constructions. The work combines theoretical coset-analysis with computational verification to produce new LCD BCH-code families with practical implications for data reliability and cryptography. It also outlines avenues for future work, such as determining the first few largest coset leaders for general $q,m$ and extending the techniques to other lengths $n=\frac{q^{m}+1}{N}$.

Abstract

BCH codes form an important subclass of cyclic codes, and are widely used in compact discs, digital audio tapes and other data storage systems to improve data reliability. As far as we know, there are few results on $q$-ary BCH codes of length $n=\frac{q^{m}+1}{q+1}$. This is because it is harder to deal with BCH codes of such length. In this paper, we study $q$-ary BCH codes with lengths $n=\frac{q^{m}+1}{q+1}$ and $n=q^m+1$. These two classes of BCH codes are always LCD codes. For $n=\frac{q^{m}+1}{q+1}$, the dimensions of narrow-sense BCH codes of length $n$ with designed distance $δ=\ell q^{\frac{m-1}{2}}+1$ are determined, where $q>2$ and $2\leq \ell \leq q-1$. Moreover, the largest coset leader is given for $m=3$ and the first two largest coset leaders are given for $q=2$. The parameters of BCH codes related to the first few largest coset leaders are investigated. Some binary BCH codes of length $n=\frac{2^m+1}{3}$ have optimal parameters. For ternary narrow-sense BCH codes of length $n=3^m+1$, a lower bound on the minimum distance of their dual codes is developed, which is good in some cases.

Theorems & Definitions (49)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• proof
• Lemma 3.1
• Lemma 3.2
• Proposition 3.1
• proof