On dually almost MRD codes

TL;DR

This paper introduces and analyzes dually AMRD codes, a class of codes nearly optimal in rank metric, providing conditions for their duality and exploring their relation to other near-MRD codes.

Contribution

It defines dually AMRD codes, establishes necessary and sufficient conditions for their existence, and relates them to codes with rank defect one and maximum 2-generalized weight.

Findings

Dually AMRD codes are the closest to MRD codes with both code and dual being almost optimal.

Necessary and sufficient conditions for dually AMRD codes are provided.

When matrix size divides the dimension, dually AMRD codes coincide with codes of rank defect one and maximum 2-generalized weight.

Abstract

In this paper we define and study a family of codes which come close to be MRD codes, so we call them AMRD codes (almost MRD). An AMRD code is a code with rank defect equal to 1. AMRD codes whose duals are AMRD are called dually AMRD. Dually AMRD codes are the closest to the MRD codes given that both they and their dual codes are almost optimal. Necessary and sufficient conditions for the codes to be dually AMRD are given. Furthermore we show that dually AMRD codes and codes of rank defect one and maximum 2-generalized weight coincide when the size of the matrix divides the dimension.