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

Let Fq be the finite field of q elements, where q=pᵐ with p being a prime number and m being a positive integer. Let C(q, n, δ, h) be a class of BCH codes of length n and designed 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^ are AMDS, then C is called a near MDS (NMDS) code. In [A class of almost MDS codes, { Finite Fields Appl.} { 79} (2022), #101996], Geng, Yang, Zhang and Zhou proved that the BCH code C(q, q+1,3,4) is an almost MDS code, where q=3ᵐ 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) 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 C(q, q+1,3,4) is a near MDS code.