Additive Complementary Dual Codes From Group Characters

Abstract

Additive codes have become an increasingly important topic in algebraic coding theory due to their applications in quantum error-correction and quantum computing. Linear Complementary Dual (LCD) codes play an important role for improving the security of information against certain attacks. Motivated by these facts, we define additive complementary dual codes (ACD for short) over a finite abelian group in terms of an arbitrary duality on the ambient space and examine their properties. We show that the best minimum weight of ACD codes is always greater than or equal to the best minimum weight of LCD codes of the same size and that this inequality is often strict. We give some matrix constructions for quaternary ACD codes from a given quaternary ACD code and also from a given binary self-orthogonal code. Moreover, we construct an algorithm to determine if a given quaternary additive code is an ACD code with respect to all possible symmetric dualities. We also determine the largest minimum distance of quaternary ACD codes for lengths <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n \leq 10$ </tex-math></inline-formula> . The obtained codes are either optimal or near optimal according to Bierbrauer <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al</i> +. (2009).