A characterization for an almost MDS code to be a near MDS code and a proof of the Geng-Yang-Zhang-Zhou conjecture

TL;DR

The paper derives a general criterion: the dual of an AMDS code is AMDS if and only if the shortened codes satisfy $\dim\mathcal{C}(I)=1$ for all $I$ with $|I|=n-k+1$, using MacWilliams identities. It then verifies this condition for the BCH code $\mathcal{C}_{(q,q+1,3,4)}$ with $q=3^m$ odd by exploiting the $\,(q+1)$-th roots of unity group and a determinant-based characterization, showing the dual is AMDS with $[q+1,4,q-3]$. Consequently, $\mathcal{C}_{(q,q+1,3,4)}$ is NMDS. This confirms the Geng–Yang–Zhang–Zhou conjecture and highlights a practical criterion to certify NMDS status via dual AMDS properties.

Abstract

Let $\mathbb{F}_q$ be the finite field of $q$ elements, where $q=p^{m}$ with $p$ being a prime number and $m$ being a positive integer. Let $\mathcal{C}_{(q, n, δ, h)}$ be a class of BCH codes of length $n$ and designed $δ$. A linear code $\mathcal{C}$ is said to be maximum distance separable (MDS) if the minimum distance $d=n-k+1$. If $d=n-k$, then $\mathcal{C}$ is called an almost MDS (AMDS) code. Moreover, if both of $\mathcal{C}$ and its dual code $\mathcal{C}^{\bot}$ are AMDS, then $\mathcal{C}$ is called a near MDS (NMDS) code. In [A class of almost MDS codes, {\it Finite Fields Appl.} {\bf 79} (2022), \#101996], Geng, Yang, Zhang and Zhou proved that the BCH code $\mathcal{C}_{(q, q+1,3,4)}$ is an almost MDS code, where $q=3^m$ and $m$ is an odd integer, and they also showed that its parameters is $[q+1, q-3, 4]$. Furthermore, they proposed a conjecture stating that the dual code $\mathcal{C}^{\bot}_{(q, q+1, 3, 4)}$ is also an AMDS code with parameters $[q+1, 4, q-3]$. In this paper, we first present a characterization for the dual code of an almost MDS code to be an almost MDS code. Then we use this result to show that the Geng-Yang-Zhang-Zhou conjecture is true. Our result together with the Geng-Yang-Zhang-Zhou theorem implies that the BCH code $\mathcal{C}_{(q, q+1,3,4)}$ is a near MDS code.

Theorems & Definitions (38)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof