Additive Complementary Pairs of Codes

TL;DR

This paper explores the algebraic structure of additive complementary pairs of codes over finite fields, providing characterizations, necessary conditions, and constructions, with applications to cryptography and code theory.

Contribution

It introduces a detailed algebraic framework for additive complementary pairs of codes, including characterizations, necessary conditions, and new construction methods.

Findings

Characterization of additive complementary pairs (ACP) using generator and parity check matrices

Identification of a necessary condition for ACP of codes

Development of constructions of ACP from LCP codes over different fields

Abstract

An additive code is an $\mathbb{F}_q$-linear subspace of $\mathbb{F}_{q^m}^n$ over $\mathbb{F}_{q^m}$, which is not a linear subspace over $\mathbb{F}_{q^m}$. Linear complementary pairs (LCP) of codes have important roles in cryptography, such as increasing the speed and capacity of digital communication and strengthening security by improving the encryption necessities to resist cryptanalytic attacks. This paper studies an algebraic structure of additive complementary pairs (ACP) of codes over $\mathbb{F}_{q^m}$. Further, we characterize an ACP of codes in analogous generator matrices and parity check matrices. Additionally, we identify a necessary condition for an ACP of codes. Besides, we present some constructions of an ACP of codes over $\mathbb{F}_{q^m}$ from LCP codes over $\mathbb{F}_{q^m}$ and also from an LCP of codes over $\mathbb{F}_{q}$. Finally, we study the constacyclic ACP of codes over $\mathbb{F}_{q^m}$ and the counting of the constacyclic ACP of codes.