Rank error-correcting pairs

TL;DR

This paper extends the concept of error-correcting pairs to rank-metric codes by introducing new vector products, establishing bounds, and demonstrating applicability to well-known rank-metric code families, generalizing classical decoding algorithms.

Contribution

It introduces new vector products that extend coordinatewise products, defines rank error-correcting pairs, and relates them to classical error-correcting pairs for the Hamming metric.

Findings

New vector products preserve polynomial symbolic products.

Rank error-correcting pairs are defined for codes over extension and base fields.

The algorithm generalizes classical error-correcting pair decoding for Hamming codes.

Abstract

Error-correcting pairs were introduced independently by Pellikaan and K\"otter as a general method of decoding linear codes with respect to the Hamming metric using coordinatewise products of vectors, and are used for many well-known families of codes. In this paper, we define new types of vector products, extending the coordinatewise product, some of which preserve symbolic products of linearized polynomials after evaluation and some of which coincide with usual products of matrices. Then we define rank error-correcting pairs for codes that are linear over the extension field and for codes that are linear over the base field, and relate both types. Bounds on the minimum rank distance of codes and MRD conditions are given. Finally we show that some well-known families of rank-metric codes admit rank error-correcting pairs, and show that the given algorithm generalizes the classical algorithm using error-correcting pairs for the Hamming metric.