On $\ell$-rank additive intersection pairs (RAIP) of codes

2026-08-11

AI Summary

This work aims to unify the characterization of additive complementary dual (ACD) codes, additive complementary pairs (ACP), and kernels of additive codes. To this end, the authors introduce a novel framework termed ℓ-rank additive intersection pairs (ℓ-RAIP), establish necessary and sufficient conditions for their existence, and demonstrate that every additive code pair is monomially equivalent to an ACP when \( q > 2 \), and every additive code is equivalent to an ACD code when \( q > 3 \). Furthermore, leveraging self-orthogonal additive codes, they propose a general construction method that yields explicit families of ℓ-RAIP codes, thereby systematically establishing broad equivalence relations between additive codes and ACD/ACP structures.

Abstract

This paper introduces and studies \(\ell\)-rank additive intersection pairs (RAIP) of codes over finite fields for a given positive integer \(\ell\). The notion of \(\ell\)-RAIP provides a common framework that generalizes additive complementary dual (ACD) codes, additive complementary pairs (ACP) of codes, and the hull of an additive code. We establish necessary and sufficient conditions characterizing when a pair of additive codes forms an \(\ell\)-RAIP. Furthermore, for (q>2), we prove that any pair of additive codes is monomially equivalent to an ACP of codes. As a consequence, for (q>3), every additive code is monomially equivalent to an ACD code. { A key contribution of the paper is a general construction method for \(\ell\)-RAIP of codes derived from self-orthogonal additive codes, which yields families of \((\ell+1)\)-RAIP of codes under suitable conditions.} In addition, several explicit constructions of \(\ell\)-RAIP of codes are presented.

Problem

• rank additive intersection pairs
• additive codes
• complementary dual codes
• self-orthogonal codes
• finite fields

Innovation

• rank additive intersection pairs
• additive complementary dual codes
• self-orthogonal additive codes
• monomial equivalence
• finite fields