Remark on subcodes of linear complementary dual codes

TL;DR

This paper proves that certain linear complementary dual codes over ternary and quaternary fields always contain subcodes of one dimension lower, leading to bounds on their minimum weights.

Contribution

It establishes the existence of specific subcodes within LCD codes and derives bounds on their minimum weights, advancing understanding of their structure.

Findings

Existence of [n,k-1] subcodes in [n,k] LCD codes

Bound on maximum minimum weights of these codes

Structural insights into ternary and quaternary LCD codes

Abstract

We show that any ternary Euclidean (resp.\ quaternary Hermitian) linear complementary dual $[n,k]$ code contains a Euclidean (resp.\ Hermitian) linear complementary dual $[n,k-1]$ subcode for $2 \le k \le n$. As a consequence, we derive a bound on the largest minimum weights among ternary Euclidean linear complementary dual codes and quaternary Hermitian linear complementary dual codes.