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Hermitian Self-dual Generalized Reed-Solomon Codes

Chun'e Zhao, Wenping Ma

TL;DR

The paper solves the long-standing problem of characterizing Hermitian self-dual generalized Reed-Solomon codes over $\mathbb{F}_{q^2}$. It proves that no such codes exist for length $n> q+1$ and, for $n\le q+1$, there are exactly two classes. It then provides two explicit construction methods corresponding to these classes, thereby completing both existence and constructive aspects. The results have practical implications for designing Hermitian self-dual MDS codes via GRS codes, with direct impact on coding theory applications where duality properties are crucial.

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.

Hermitian Self-dual Generalized Reed-Solomon Codes

TL;DR

The paper solves the long-standing problem of characterizing Hermitian self-dual generalized Reed-Solomon codes over . It proves that no such codes exist for length and, for , there are exactly two classes. It then provides two explicit construction methods corresponding to these classes, thereby completing both existence and constructive aspects. The results have practical implications for designing Hermitian self-dual MDS codes via GRS codes, with direct impact on coding theory applications where duality properties are crucial.

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 , this paper is devoted to an investigation of GRS codes in the case where . First, we prove that when , 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.
Paper Structure (9 sections, 10 theorems, 32 equations)

This paper contains 9 sections, 10 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hermitian Self-Dual Codes
  4. GRS Codes
  5. Hermitian Self-dual GRS codes
  6. Constructions
  7. The First class of Hermitian self-dual GRS codes
  8. The Second class of Hermitian self-dual GRS codes
  9. Conclusion

Key Result

Theorem 1

Let $\alpha_{1},\alpha_{2},\cdots,\alpha_{n}$ be distinct elements in the finite field $\mathbb{F}_{q^{2}}$, where $q$ is a prime power. Let $G(x)=\prod\limits_{i=1}^{n}(x-\alpha_{i})=x^{n}-\sum\limits_{i=0}^{n-1}c_{i}x^{i}$, then the matrix has $\alpha_{1},\alpha_{2},\cdots,\alpha_{n}$ as its eigenvalues, and the eigenvector corresponding to each $\alpha_{i}$ is $(1,\alpha_{i},\alpha_{i}^{2},\cd

Theorems & Definitions (20)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Corollary 1
  • proof
  • Lemma 1
  • proof
  • Theorem 3
  • proof
  • ...and 10 more