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Neighbors, neighbor graphs and invariant rings in coding theory

Himadri Shekhar Chakraborty, Williams Chiari, Tsuyoshi Miezaki, Manabu Oura

Abstract

In the present paper, we discuss the class of Type III and Type IV codes from the perspectives of neighbors. Our investigation analogously extends the results originally presented by Dougherty [8] concerning the neighbor graph of binary self-dual codes. Moreover, as an application of neighbors in invariant theory, we show that the ring of the weight enumerators of Type II code $d_{n}^{+}$ and its neighbors in arbitrary genus is finitely generated. Finally, we obtain a minimal set of generators of this ring up to the space of degree 24 and genus 3.

Neighbors, neighbor graphs and invariant rings in coding theory

Abstract

In the present paper, we discuss the class of Type III and Type IV codes from the perspectives of neighbors. Our investigation analogously extends the results originally presented by Dougherty [8] concerning the neighbor graph of binary self-dual codes. Moreover, as an application of neighbors in invariant theory, we show that the ring of the weight enumerators of Type II code and its neighbors in arbitrary genus is finitely generated. Finally, we obtain a minimal set of generators of this ring up to the space of degree 24 and genus 3.

Paper Structure

This paper contains 9 sections, 43 theorems, 53 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Linear codes
  4. Graphs
  5. Type $\mathrm{III}$ codes
  6. Type $\mathrm{III}$ code with all-ones vector
  7. Type $\mathrm{IV}$ codes
  8. $k$-Neighbor graphs
  9. An application of neighbors in invariant theory

Key Result

Lemma 2.1

Let $n \equiv 0\pmod 4$. Then the weight of any self-orthogonal vector in $\mathbb{F}_{3}^{n}$ is divisible by $3$.

Theorems & Definitions (104)

  • Definition 1.1
  • Definition 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 3.1
  • proof
  • ...and 94 more