Characterizations of NMDS codes and a proof of the Geng-Yang-Zhang-Zhou conjecture

Let $F_q$ be the finite field of q elements, where $q=p^m$ with p being a prime number and m being a positive integer. Let $C_{(q,n,\delta ,h)}$ be a class of BCH codes of length n and designed distance δ. A linear code $C$ is said to be maximum distance separable (MDS) if the minimum distance $d=n-k+1$. If $d=n-k$, then $C$ is called an almost MDS (AMDS) code. Moreover, if both of $C$ and its dual code $C^{\perp }$ are AMDS, then $C$ is called a near MDS (NMDS) code. In [9], Geng, Yang, Zhang and Zhou proved that the BCH code $C_{(q,q+1,3,4)}$ 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 $C_{(q,q+1,3,4)}^{\perp }$ is also an AMDS code with parameters $[q+1,4,q-3]$. In this paper, we introduce the concept of subset code and use it together with the MacWilliams identity to establish characterizations for the dual code of an AMDS code to be an AMDS code. Then by this criteria, we 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 $C_{(q,q+1,3,4)}$ is an NMDS code.