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Construction of MDS Euclidean Self-Dual Codes via Multiple Subsets

Weirong Meng, Weijun Fang, Fang-Wei Fu, Haiyan Zhou, Ziyi Gu

TL;DR

The paper tackles the problem of constructing $q$-ary MDS Euclidean self-dual codes of various even lengths by leveraging (extended) generalized Reed-Solomon codes and carefully engineered evaluation sets. The authors introduce a general method to select evaluation sets from multiple finite-field subsets (S from unions and intersections of A_i), with a central result (Theorem 1) ensuring the self-duality criteria hold under quadratic-character constraints. Building on this, they present six new code families derived from norm-function fibers and from unions of multiplicative subgroups and their cosets, achieving an unprecedented coverage of lengths (over 85% of all possibilities). The work also provides detailed comparisons to prior constructions and offers concrete numerical examples, underscoring the practical impact of expanding the spectrum of length parameters for MDS Euclidean self-dual codes. Overall, the paper advances the state of the art by broadening the attainable lengths and providing a flexible framework for evaluating-sets that guarantee Euclidean self-duality in GRS-based codes.

Abstract

MDS self-dual codes have good algebraic structure, and their parameters are completely determined by the code length. In recent years, the construction of MDS Euclidean self-dual codes with new lengths has become an important issue in coding theory. In this paper, we are committed to constructing new MDS Euclidean self-dual codes via generalized Reed-Solomon (GRS) codes and their extended (EGRS) codes. The main effort of our constructions is to find suitable subsets of finite fields as the evaluation sets, ensuring that the corresponding (extended) GRS codes are Euclidean self-dual. Firstly, we present a method for selecting evaluation sets from multiple intersecting subsets and provide a theorem to guarantee that the chosen evaluation sets meet the desired criteria. Secondly, based on this theorem, we construct six new classes of MDS Euclidean self-dual codes using the norm function, as well as the union of three multiplicity subgroups and their cosets respectively. Finally, in our constructions, the proportion of possible MDS Euclidean self-dual codes exceeds 85\%, which is much higher than previously reported results.

Construction of MDS Euclidean Self-Dual Codes via Multiple Subsets

TL;DR

The paper tackles the problem of constructing -ary MDS Euclidean self-dual codes of various even lengths by leveraging (extended) generalized Reed-Solomon codes and carefully engineered evaluation sets. The authors introduce a general method to select evaluation sets from multiple finite-field subsets (S from unions and intersections of A_i), with a central result (Theorem 1) ensuring the self-duality criteria hold under quadratic-character constraints. Building on this, they present six new code families derived from norm-function fibers and from unions of multiplicative subgroups and their cosets, achieving an unprecedented coverage of lengths (over 85% of all possibilities). The work also provides detailed comparisons to prior constructions and offers concrete numerical examples, underscoring the practical impact of expanding the spectrum of length parameters for MDS Euclidean self-dual codes. Overall, the paper advances the state of the art by broadening the attainable lengths and providing a flexible framework for evaluating-sets that guarantee Euclidean self-duality in GRS-based codes.

Abstract

MDS self-dual codes have good algebraic structure, and their parameters are completely determined by the code length. In recent years, the construction of MDS Euclidean self-dual codes with new lengths has become an important issue in coding theory. In this paper, we are committed to constructing new MDS Euclidean self-dual codes via generalized Reed-Solomon (GRS) codes and their extended (EGRS) codes. The main effort of our constructions is to find suitable subsets of finite fields as the evaluation sets, ensuring that the corresponding (extended) GRS codes are Euclidean self-dual. Firstly, we present a method for selecting evaluation sets from multiple intersecting subsets and provide a theorem to guarantee that the chosen evaluation sets meet the desired criteria. Secondly, based on this theorem, we construct six new classes of MDS Euclidean self-dual codes using the norm function, as well as the union of three multiplicity subgroups and their cosets respectively. Finally, in our constructions, the proportion of possible MDS Euclidean self-dual codes exceeds 85\%, which is much higher than previously reported results.

Paper Structure

This paper contains 13 sections, 19 theorems, 165 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Evaluation sets from multiple sets of finite fields
  4. MDS Euclidean self-dual codes from norm functions
  5. MDS Euclidean self-dual codes from multipliacative subgroups
  6. The special case of three sets
  7. The general case of three sets
  8. Comparisons
  9. Conclusion
  10. The proof of Theorem 3
  11. The proof of Corollary 1
  12. The proof of Corollary 3
  13. Some known results on MDS Euclidean self-dual codes of length $n$

Key Result

Lemma 1

Let $E=\bigcup\limits_{i=1}^{n}E_{i}$, where $E_{1},E_{2},\dots, E_{n}$ are $n$ pairwise disjoint subsets of $E$, then for any $e\in E_i$, we have

Figures (2)

  • Figure 1: Two sets.
  • Figure 2: Three sets.

Theorems & Definitions (46)

  • Lemma 1: FXF21
  • Lemma 2: ZF20
  • Lemma 3: ZF20
  • Example 1
  • Example 2
  • Lemma 4
  • proof
  • Theorem 1
  • proof
  • Remark 1
  • ...and 36 more