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

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 $\mathcal{C}_{(q, n, \delta, h)}$ be a class of BCH codes of length $n$ and designed $\delta$. 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.
Comment: 25 pages. One reference was added and several typos were corrected in this version