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A Survey on Codes from Simplicial Complexes

Yansheng Wu, Chao Li, Jong Yoon Hyun

Abstract

In the field of mathematics, a purely combinatorial equivalent to a simplicial complex, or more generally, a down-set, is an abstract structure known as a family of sets. This family is closed under the operation of taking subsets, meaning that every subset of a set within the family is also included in the family. The purpose of this paper is two-fold. Firstly, it aims to present a comprehensive survey of recent results in the field. This survey intends to provide an overview of the advancements made in codes constructed from simplicial complexes. Secondly, the paper seeks to propose open problems that are anticipated to stimulate further research in this area. By highlighting these open problems, the paper aims to encourage and inspire future investigations and developments in the field of codes derived from simplicial complexes.

A Survey on Codes from Simplicial Complexes

Abstract

In the field of mathematics, a purely combinatorial equivalent to a simplicial complex, or more generally, a down-set, is an abstract structure known as a family of sets. This family is closed under the operation of taking subsets, meaning that every subset of a set within the family is also included in the family. The purpose of this paper is two-fold. Firstly, it aims to present a comprehensive survey of recent results in the field. This survey intends to provide an overview of the advancements made in codes constructed from simplicial complexes. Secondly, the paper seeks to propose open problems that are anticipated to stimulate further research in this area. By highlighting these open problems, the paper aims to encourage and inspire future investigations and developments in the field of codes derived from simplicial complexes.
Paper Structure (27 sections, 56 theorems, 36 equations, 1 table)

This paper contains 27 sections, 56 theorems, 36 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Two Constructions of Linear Codes
  4. Three Different Metrics for Codes
  5. Minimal Linear Codes
  6. Subfield Codes
  7. The Projective Solomon-Stiffler Codes
  8. Two Bounds of Linear Codes
  9. Simplicial Complexes and Posets
  10. Codes over finite rings
  11. Codes over ring $\mathbb Z_4$
  12. Codes over the ring $\Bbb F_p + u\Bbb F_p$ with $u^2=0$
  13. Codes over ring $\Bbb F_p + u\Bbb F_p$ with $u^2=u$
  14. Codes over rings $\Bbb F_2 + u\Bbb F_2 + v\Bbb F_2 + uv \Bbb F_2$, where $u^2 = 0$, $v^2 = 0$, and $uv = vu$
  15. $\Bbb Z_p\Bbb Z_p[u]$-Additive codes
  16. ...and 12 more sections

Key Result

Lemma 1

(Ashikhmin-Barg AB) Let $\mathcal{C}$ be a linear code over $\mathbb{F}_q$ with $\hbox{wt}_{min}$ and $\hbox{wt}_{max}$ as minimum and maximum weights, respectively, of its non-zero codewords. If then $\mathcal{C}$ is minimal.

Theorems & Definitions (56)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Theorem 9
  • Theorem 10
  • ...and 46 more