New Characterization and Parametrization of LCD Codes

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 orthogonal or symplectic basis. Using such a characterization, we solve a conjecture proposed by Galvez <italic>et al.</italic> 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 on 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 <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula>-ary LCD codes have orthonormal basis, where <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> is a power of an odd prime.