Hermitian Self-dual Generalized Reed-Solomon Codes

TL;DR

This paper fully characterizes Hermitian self-dual GRS codes for lengths up to q+1, proving only two classes exist and providing explicit construction methods, thus solving their existence and construction problem.

Contribution

It proves that only two classes of Hermitian self-dual GRS codes exist for lengths up to q+1 and introduces two explicit construction methods.

Findings

Only two classes of Hermitian self-dual GRS codes exist for n ≤ q+1.

The paper provides explicit construction methods for these codes.

The conjecture in [13] is confirmed and proved.

Abstract

Maximum Distance Separable (MDS) self-dual codes are of significant theoretical and practical importance. Generalized Reed-Solomon (GRS) codes are the most prominent MDS codes. Correspondingly there have been many research on constructions of Euclidean self-dual MDS codes by using GRS codes. However, the study on Hermitian self-dual GRS codes is relatively limited. Since Hermitian self-dual GRS codes do not exist for $n>q+1$, this paper is devoted to an investigation of GRS codes in the case where $n\le q+1$. First, we prove that when $n\leq q+1$, there are only two classes of Hermitian self-dual GRS codes, confirming the conjecture in [13] and providing its proof simultaneously. Second, we present two explicit construction methods. Thus, the existence and construction of Hermitian self-dual GRS codes are fully solved.