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Infinite families of optimal and minimal codes over rings using simplicial complexes

Yanan Wu, Tingting Pang, Nian Li, Yanbin Pan, Xiangyong Zeng

TL;DR

The paper addresses constructing infinite families of linear codes over extensions $\mathcal{R}=E^s$ or $F^s$ of non-unital non-commutative rings using general simplicial complexes, and analyzes their Lee-weight, Gray-map images, and subfield-like codes. It introduces a universal parameter-determination method based on inner-product orthogonality and inclusion-exclusion, and derives explicit Lee-weight distributions when the defining simplicial complex is generated by a single maximal element. The work delivers numerous $\,\mathbb{F}_q$-codes with self-orthogonality and minimality properties, as well as distance-optimal families with respect to the Griesmer bound, and establishes conditions for Hermitian self-orthogonality when $q=4$. Overall, it broadens the scope of simplicial-complex-based code constructions to extension rings, yielding flexible, high-performance code families with potential applications in secret sharing and secure computation.

Abstract

In this paper, several infinite families of codes over the extension of non-unital non-commutative rings are constructed utilizing general simplicial complexes. Thanks to the special structure of the defining sets, the principal parameters of these codes are characterized. Specially, when the employed simplicial complexes are generated by a single maximal element, we determine their Lee weight distributions completely. Furthermore, by considering the Gray image codes and the corresponding subfield-like codes, numerous of linear codes over $\mathbb{F}_q$ are also obtained, where $q$ is a prime power. Certain conditions are given to ensure the above linear codes are (Hermitian) self-orthogonal in the case of $q=2,3,4$. It is noteworthy that most of the derived codes over $\mathbb{F}_q$ satisfy the Ashikhmin-Barg's condition for minimality. Besides, we obtain two infinite families of distance-optimal codes over $\mathbb{F}_q$ with respect to the Griesmer bound.

Infinite families of optimal and minimal codes over rings using simplicial complexes

TL;DR

The paper addresses constructing infinite families of linear codes over extensions or of non-unital non-commutative rings using general simplicial complexes, and analyzes their Lee-weight, Gray-map images, and subfield-like codes. It introduces a universal parameter-determination method based on inner-product orthogonality and inclusion-exclusion, and derives explicit Lee-weight distributions when the defining simplicial complex is generated by a single maximal element. The work delivers numerous -codes with self-orthogonality and minimality properties, as well as distance-optimal families with respect to the Griesmer bound, and establishes conditions for Hermitian self-orthogonality when . Overall, it broadens the scope of simplicial-complex-based code constructions to extension rings, yielding flexible, high-performance code families with potential applications in secret sharing and secure computation.

Abstract

In this paper, several infinite families of codes over the extension of non-unital non-commutative rings are constructed utilizing general simplicial complexes. Thanks to the special structure of the defining sets, the principal parameters of these codes are characterized. Specially, when the employed simplicial complexes are generated by a single maximal element, we determine their Lee weight distributions completely. Furthermore, by considering the Gray image codes and the corresponding subfield-like codes, numerous of linear codes over are also obtained, where is a prime power. Certain conditions are given to ensure the above linear codes are (Hermitian) self-orthogonal in the case of . It is noteworthy that most of the derived codes over satisfy the Ashikhmin-Barg's condition for minimality. Besides, we obtain two infinite families of distance-optimal codes over with respect to the Griesmer bound.
Paper Structure (18 sections, 23 theorems, 91 equations, 10 tables)

This paper contains 18 sections, 23 theorems, 91 equations, 10 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Motivation
  4. Our Contributions
  5. Definitions and Preliminaries
  6. Lee Weight and Gray Map
  7. The Generic Construction of Linear Codes
  8. Simplicial Complexes
  9. Our Construction and the Calculation of Lee Weight
  10. Useful Auxiliary Lemmas
  11. Linear Codes over Non-unital Non-commutative Rings
  12. Codes over Rings and the Corresponding Subfield-like Codes
  13. Explicit Lee Weight Distribution of $\mathcal{C}_L$ When $l=k=1$
  14. Structures of the Gray Image Codes and Subfield-like Codes
  15. Self-orthogonal Codes
  16. ...and 3 more sections

Key Result

Lemma 1

Let $\Delta$ be a simplicial complex of $\mathbb{F}_q^m$ with the set of maximal elements $\mathcal{F}= \{F_1,\,\dots,\,F_l \}$. Then the size of $\Delta$ is where $\cap S=\bigcap\limits_{F\in S}F$.

Theorems & Definitions (28)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Theorem 1
  • ...and 18 more