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=p^m with p being a prime number and m being a positive integer. Let C (ₐ, ₍, , ₇) be a class of BCH codes of length n and designed. 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 (ₐ, ₐ+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^ (ₐ, ₐ+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 (ₐ, ₐ+₁, ₃, ₄) is a near MDS code.