On the existence of quaternary Hermitian LCD codes with Hermitian dual distance $1$

TL;DR

This paper investigates the existence conditions of quaternary Hermitian LCD codes, establishing nonexistence results for codes with certain minimum distances and dual distances, thereby extending previous nonexistence theorems.

Contribution

It generalizes prior nonexistence results for quaternary Hermitian LCD codes, linking the absence of codes with dual distance ≥ 2 to the absence of codes with dual distance 1.

Findings

No quaternary Hermitian LCD codes with dual distance 1 exist if none with dual distance ≥ 2 exist for the same parameters.

Generalizes a known nonexistence result by Araya, Harada, and Saito.

Provides a broader understanding of the parameter constraints for Hermitian LCD codes.

Abstract

For $k \ge 2$ and a positive integer $d_0$, we show that if there exists no quaternary Hermitian linear complementary dual $[n,k,d]$ code with $d \ge d_0$ and Hermitian dual distance greater than or equal to $2$, then there exists no quaternary Hermitian linear complementary dual $[n,k,d]$ code with $d \ge d_0$ and Hermitian dual distance $1$. As a consequence, we generalize a result by Araya, Harada and Saito on the nonexistence of some quaternary Hermitian linear complementary dual codes.