Several constructions of optimal LCD codes over small finite fields

TL;DR

This paper introduces new methods for constructing optimal LCD codes over small finite fields, improving bounds and disproving a recent conjecture, with practical implications in data security.

Contribution

The paper develops modified construction methods for LCD codes over small fields, expanding the class of known codes and challenging existing conjectures.

Findings

All odd-like binary LCD codes can be constructed with the new methods.

The largest minimum distances of LCD codes are improved.

Two counterexamples disprove Bouyuklieva's conjecture.

Abstract

Linear complementary dual (LCD) codes are linear codes which intersect their dual codes trivially, which have been of interest and extensively studied due to their practical applications in computational complexity and information protection. In this paper, we give some methods for constructing LCD codes over small finite fields by modifying some typical methods for constructing linear codes. We show that all odd-like binary LCD codes, ternary LCD codes and quaternary Hermitian LCD codes can be constructed using the modified methods. Our results improve the known lower bounds on the largest minimum distances of LCD codes. Furthermore, we give two counterexamples to disprove the conjecture proposed by Bouyuklieva (Des. Codes Cryptogr. 89(11): 2445-2461, 2021).