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Shortest LCD embeddings of binary, ternary and quaternary linear codes

Junmin An, Ji-Hoon Hong, Jon-Lark Kim, Haeun Lim

TL;DR

The paper addresses how to embed a given binary, ternary, or quaternary linear code into an LCD code with minimal length. It shows that the minimal additional length equals the hull dimension $\ell=\dim(\mathrm{Hull}(\mathcal{C}))$, so the shortest LCD embedding has length $n+\ell$, and provides explicit constructions via hull and LCD parts to guarantee the LCD property. By applying the method to Hamming codes, the authors produce new optimal LCD codes with improved parameters, including ternary $[23,4,14]$, $[23,5,12]$, $[24,6,12]$, $[25,5,14]$, and quaternary $[21,10,8]$, highlighting the practical utility of their approach for constructing high-distance LCD codes across fields. The results advance the design of LCD codes for applications in cryptography and secure decoding by providing a principled embedding technique and concrete new codes.

Abstract

In the recent years, there has been active research on self-orthogonal embeddings of linear codes since they yielded some optimal self-orthogonal codes. LCD codes have a trivial hull so they are counterparts of self-orthogonal codes. So it is a natural question whether one can embed linear codes into optimal LCD codes. To answer it, we first determine the number of columns to be added to a generator matrix of a linear code in order to embed the given code into an LCD code. Then we characterize all possible forms of shortest LCD embeddings of a linear code. As examples, we start from binary and ternary Hamming codes of small lengths and obtain optimal LCD codes with minimum distance 4. Furthermore, we find new ternary LCD codes with parameters including $[23, 4, 14]$, $[23, 5, 12]$, $[24, 6, 12]$, and $[25, 5, 14]$ and a new quaternary LCD $[21, 10, 8]$ code, each of which has minimum distance one greater than those of known codes. This shows that our shortest LCD embedding method is useful in finding optimal LCD codes over various fields.

Shortest LCD embeddings of binary, ternary and quaternary linear codes

TL;DR

The paper addresses how to embed a given binary, ternary, or quaternary linear code into an LCD code with minimal length. It shows that the minimal additional length equals the hull dimension , so the shortest LCD embedding has length , and provides explicit constructions via hull and LCD parts to guarantee the LCD property. By applying the method to Hamming codes, the authors produce new optimal LCD codes with improved parameters, including ternary , , , , and quaternary , highlighting the practical utility of their approach for constructing high-distance LCD codes across fields. The results advance the design of LCD codes for applications in cryptography and secure decoding by providing a principled embedding technique and concrete new codes.

Abstract

In the recent years, there has been active research on self-orthogonal embeddings of linear codes since they yielded some optimal self-orthogonal codes. LCD codes have a trivial hull so they are counterparts of self-orthogonal codes. So it is a natural question whether one can embed linear codes into optimal LCD codes. To answer it, we first determine the number of columns to be added to a generator matrix of a linear code in order to embed the given code into an LCD code. Then we characterize all possible forms of shortest LCD embeddings of a linear code. As examples, we start from binary and ternary Hamming codes of small lengths and obtain optimal LCD codes with minimum distance 4. Furthermore, we find new ternary LCD codes with parameters including , , , and and a new quaternary LCD code, each of which has minimum distance one greater than those of known codes. This shows that our shortest LCD embedding method is useful in finding optimal LCD codes over various fields.
Paper Structure (6 sections, 13 theorems, 28 equations, 1 table)

This paper contains 6 sections, 13 theorems, 28 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Shortest LCD embeddings of codes
  4. Optimal LCD codes
  5. Conclusion
  6. Acknowledgement

Key Result

Theorem 2.1

Let $\mathcal{C}$ be an $[n, k]$ linear code over $\mathbb{F}_q$ with generator matrix $G$. Then the dimension $\ell$ of the Euclidean hull ${\rm Hull}(\mathcal{C})$ is given by where $G^T$ is the transpose of $G$. Furthermore, for a linear $[n, k]$ code $\mathcal{C}$ over $\mathbb{F}_{q^2}$, the dimension $\ell_h$ of Hermitian hull is given by where $\sigma(G)$ is the matrix obtained by taking

Theorems & Definitions (25)

  • Theorem 2.1: Li-hull-dim
  • Definition 3.1
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • proof
  • Remark 1
  • Lemma 3.4
  • ...and 15 more