The dual codes of two classes of LCD BCH codes

TL;DR

This work addresses the duals of two LCD BCH code families: narrow-sense cyclic BCH codes of length $n=q^m+1$ and narrow-sense negacyclic BCH codes of length $n=(q^m+1)/2$ over $\mathbb{F}_q$ with $q$ an odd prime power and $q\equiv 3\pmod{4}$, respectively. It develops lower bounds on the minimum distances of the duals, and introduces the notion of dually-BCH codes for both cyclic and negacyclic settings, providing necessary and sufficient conditions in terms of designed distances for when the duals remain BCH codes (cyclic or negacyclic) and when even-like subcodes are cyclic dually-BCH. The results include explicit characterizations in terms of coset leaders ($\delta_1,\delta_2,\phi_i$) and the defining distance $\delta$, with several theorems (e.g., $t4.1$–$t4.3$, $t3.1$–$t3.5$) supported by coset-structure lemmas. These findings yield practical lower-distance guarantees and enhance the design space for LCD/dually-BCH codes, while outlining remaining cases not covered (such as $m\equiv 0 \pmod{4}$ in certain settings).

Abstract

Cyclic BCH codes and negacyclic BCH codes form important subclasses of cyclic codes and negacyclic codes, respectively, and can produce optimal linear codes in many cases. To the best of our knowledge, there are few results on the dual codes of cyclic and negacyclic BCH codes. In this paper, we study the dual codes of narrow-sense cyclic BCH codes of length $q^m+1$ over a finite field $\mathbb{F}_q$, where $q$ is an odd prime power, and the dual codes of narrow-sense negacyclic BCH codes of length $\frac{q^{m}+1}{2}$ over $\mathbb{F}_q$, where $q$ is an odd prime power satisfying $q\equiv 3~({\rm mod}~4)$. Some lower bounds on the minimum distances of the dual codes are established, which are very close to the true minimum distances of the dual codes in many cases. Sufficient and necessary conditions for the even-like subcodes of narrow-sense cyclic BCH codes of length $q^{m}+1$ being cyclic dually-BCH codes are given in terms of designed distances, where $q$ is odd and $m$ is odd or $m\equiv 2~({\rm mod~}4)$. The concept of negacyclic dually-BCH codes is proposed, and sufficient and necessary conditions in terms of designed distances are presented to ensure that narrow-sense negacyclic BCH codes of length $\frac{q^{m}+1}{2}$ are dually-BCH codes, where $q\equiv 3~({\rm mod}~4)$.

Theorems & Definitions (65)

• Lemma 2.1
• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 3.1
• proof
• Theorem 3.1
• proof
• Example 3.1
• Example 3.2