Shortest LCD embeddings of binary, ternary and quaternary linear codes

TL;DR

This paper investigates the shortest embeddings of linear codes into LCD codes, characterizes their forms, and constructs new optimal LCD codes over binary, ternary, and quaternary fields with improved parameters.

Contribution

It provides a method to embed linear codes into shortest LCD codes and constructs new optimal LCD codes with better parameters than previously known.

Findings

Determined the number of columns needed for embedding linear codes into LCD codes.

Characterized all possible shortest LCD embeddings of linear codes.

Constructed new optimal LCD codes with improved minimum distances.

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.