New characterization and parametrization of LCD Codes

TL;DR

This paper introduces a new way to characterize binary LCD codes using symplectic bases, solves a conjecture on their minimum distance, and analyzes their symmetry properties under the orthogonal group.

Contribution

It provides a novel characterization of binary LCD codes, solves an open conjecture, and explores the structure and orbit sizes of LCD codes under group actions.

Findings

Solved the conjecture on minimum distance of binary LCD codes.

Most binary LCD codes are odd-like with odd-like duals.

Approximately half of q-ary LCD codes have orthonormal bases.

Abstract

Linear complementary dual (LCD) cyclic codes were referred historically to as reversible cyclic codes, which had applications in data storage. Due to a newly discovered application in cryptography, there has been renewed interest in LCD codes. In particular, it has been shown that binary LCD codes play an important role in implementations against side-channel attacks and fault injection attacks. In this paper, we first present a new characterization of binary LCD codes in terms of their symplectic basis. Using such a characterization,we solve a conjecture proposed by Galvez et al. on the minimum distance of binary LCD codes. Next, we consider the action of the orthogonal group on the set of all LCD codes, determine all possible orbits of this action, derive simple closed formulas of the size of the orbits, and present some asymptotic results of the size of the corresponding orbits. Our results show that almost all binary LCD codes are odd-like codes with odd-like duals, and about half of q-ary LCD codes have orthonormal basis, where q is a power of an odd prime.