LCD codes over ${\mathbb F}_q $ are as good as linear codes for q at least four

TL;DR

This paper proves that over finite fields with q at least four, every linear code is monomial equivalent to an LCD code, making LCD codes as good as linear codes for these fields, including optimal and MDS codes.

Contribution

It establishes that for q ≥ 4, all linear codes are monomial equivalent to LCD codes, extending the equivalence to optimal and algebraic geometric codes.

Findings

Every F_q-linear code is monomial equivalent to an LCD code for q ≥ 4.

Existence of LCD codes with parameters matching optimal and MDS codes.

LCD codes can surpass the Gilbert-Varshamov bound for q ≥ 49.

Abstract

The hull $H(C)$ of a linear code $C$ is defined by $H(C)=C \cap C^\perp$. A linear code with a complementary dual (LCD) is a linear code with $H(C)=\{0\}$. The dimension of the hull of a code is an invariant under permutation equivalence. For binary and ternary codes the dimension of the hull is also invariant under monomial equivalence and we show that this invariant is determined by the extended weight enumerator of the code.\\ The hull of a code is not invariant under monomial equivalence if $q\geq 4$. We show that every ${\mathbb F}_q $-linear code is monomial equivalent with an LCD code in case $q \geq 4$. The proof uses techniques from Gr\"obner basis theory. We conclude that if there exists an ${\mathbb F}_q $-linear code with parameters $[n,k,d]_q$ and $q \geq 4$, then there exists also a LCD code with the same parameters. Hence this holds for optimal and MDS codes. In particular there exist LCD codes that are above the Gilbert-Varshamov bound if $q$ is a square and $q\geq 49$ by the existence of such codes that are algebraic geometric.\\ Similar results are obtained with respect to Hermitian LCD codes.