Shortest LCD embeddings of binary, ternary and quaternary linear codes

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

doi:10.3934/amc.2026048

Article Contents

2026, Volume 25: 192-202. Doi: 10.3934/amc.2026048

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Shortest LCD embeddings of binary, ternary and quaternary linear codes

1.
Department of Mathematics, Sogang University, Seoul, Korea
2.
Institute for Mathematical and Data Sciences, Sogang University, Seoul, Korea

^*Corresponding author: Junmin An
^*Corresponding author: Junmin An

Received: January 25, 2025

Revised: April 23, 2026

Published online: May 28, 2026

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Abstract

In recent years, there has been active research on self-orthogonal embeddings of linear codes since they have yielded some optimal self-orthogonal codes. LCD codes have a trivial hull, so they are counterparts of self-orthogonal codes. It is therefore a natural question whether one can embed linear codes into distance-optimal LCD codes. To answer this, 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 the shortest LCD embeddings of a linear code. Using the shortest LCD embedding method, 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 the known codes. This shows that our shortest LCD embedding method is useful in finding distance-optimal LCD codes over various fields.

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Mathematics Subject Classification: Primary: 94B05.

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Table 1. New ternary and quaternary LCD codes

Code	$ q $	New parameters	BKLC	$ \ell $	Known parameters
$ \mathcal{C}_{3, 1} $	3	$ [23, 4, {\bf{14}}] $	$ [19, 4, 12] $	4	$ [23, 4, 13] $ [18]
$ \mathcal{C}_{3, 2} $	3	$ [23, 5, {\bf{12}}] $	$ [19, 5, 11] $	4	$ [23, 5, 11] $ [18]
$ \mathcal{C}_{3, 3} $	3	$ [24, 6, {\bf{12}}] $	$ [20, 6, 10] $	4	$ [24, 6, 11] $ [18]
$ \mathcal{C}_{3, 4} $	3	$ [25, 5, {\bf{14}}] $	$ [20, 5, 12] $	5	$ [25, 5, 13] $ [18]
$ \mathcal{C}_{4, 1} $	4	$ [21, 10, {\bf{8}}] $	$ [20, 10, 8] $	1	$ [21, 10, 7] $ [21]

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