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Orthogonal projectors of LCD codes and their graph representations

Keita Ishizuka

TL;DR

The work establishes a structural bridge between LCD codes and graph theory by representing LCD codes via their orthogonal projector $A$ with $A^T=A$ and $A^2=A$. It proves a one-to-one correspondence between binary even LCD codes and certain simple graphs, and between ternary LCD codes and certain two-graphs, with code equivalence corresponding to graph isomorphism or switching. It then derives minimum-weight bounds for codes arising from strongly regular graphs and demonstrates applications, including a Paley graph example that attains the largest known minimum weight within its SRG class. The results enable transferring graph-theoretic insights to LCD-code construction and analysis, offering new avenues for code optimization and understanding via combinatorial structures.

Abstract

We establish one-to-one correspondences between (i) binary even LCD codes and certain simple graphs, and (ii) ternary LCD codes and certain two-graphs.

Orthogonal projectors of LCD codes and their graph representations

TL;DR

The work establishes a structural bridge between LCD codes and graph theory by representing LCD codes via their orthogonal projector with and . It proves a one-to-one correspondence between binary even LCD codes and certain simple graphs, and between ternary LCD codes and certain two-graphs, with code equivalence corresponding to graph isomorphism or switching. It then derives minimum-weight bounds for codes arising from strongly regular graphs and demonstrates applications, including a Paley graph example that attains the largest known minimum weight within its SRG class. The results enable transferring graph-theoretic insights to LCD-code construction and analysis, offering new avenues for code optimization and understanding via combinatorial structures.

Abstract

We establish one-to-one correspondences between (i) binary even LCD codes and certain simple graphs, and (ii) ternary LCD codes and certain two-graphs.
Paper Structure (12 sections, 17 theorems, 6 equations)

This paper contains 12 sections, 17 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. LCD codes
  4. Strongly regular graphs
  5. Two-graphs
  6. Main results
  7. Binary LCD codes and simple graphs
  8. Ternary LCD codes and two-graphs
  9. Bounds on minimum weights
  10. Applications
  11. Concluding remarks
  12. Acknowledgement

Key Result

Theorem 2.1

Let $C$ be an $[n,k]$ code over $\mathbb{F}_q$ and let $G$ be a generator matrix of $C$. Then $C$ is an LCD code if and only if the $k \times k$ matrix $GG^T$ is nonsingular. Moreover, if $C$ is an LCD code, then $\Pi_c=G^T(GG^T)^{-1}G$ is the orthogonal projector from $\mathbb{F}_q^n$ to $C$.

Theorems & Definitions (33)

  • Theorem 2.1: Proposition 1 massey_linear_1992
  • Theorem 2.2: Lemma 2 bouyuklieva_optimal_2021
  • Theorem 2.3: Proposition 2 bouyuklieva_optimal_2021
  • Theorem 2.4
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 3.3
  • Lemma 3.4
  • ...and 23 more