Additive complementary dual codes over $\F_4$

TL;DR

This paper investigates additive complementary dual codes over 4, exploring their properties, constructions from binary codes, and relevance to quantum codes, expanding the understanding of LCD-like codes in additive settings.

Contribution

It introduces the concept of additive complementary dual (ACD) codes over 4, providing new constructions and analyzing their properties, especially in relation to quantum code applications.

Findings

Constructed ACD codes from binary codes using trace Hermitian and Euclidean inner products.

Analyzed properties of ACD codes and their relation to quantum codes.

Extended techniques from LCD code studies to additive code context.

Abstract

A linear code is linear complementary dual (LCD) if it meets its dual trivially. LCD codes have been a hot topic recently due to Boolean masking application in the security of embarked electronics (Carlet and Guilley, 2014). Additive codes over $\F_4$ are $\F_4$-codes that are stable by codeword addition but not necessarily by scalar multiplication. An additive code over $\F_4$ is additive complementary dual (ACD) if it meets its dual trivially. The aim of this research is to study such codes which meet their dual trivially. All the techniques and problems used to study LCD codes are potentially relevant to ACD codes. Interesting constructions of ACD codes from binary codes are given with respect to the trace Hermitian and trace Euclidean inner product. The former product is relevant to quantum codes.