Recursive decoding of binary rank Reed-Muller codes and Plotkin construction for matrix codes

TL;DR

This paper develops a recursive decoding algorithm for binary rank metric Reed-Muller codes inspired by classical Reed-Muller decoding, achieving better asymptotic complexity for certain subclasses and introducing a rank metric Plotkin construction for matrix codes.

Contribution

It introduces a novel recursive decoding method for binary rank metric Reed-Muller codes and a rank metric Plotkin construction with associated decoding, improving efficiency over existing algorithms.

Findings

Recursive decoding algorithm outperforms previous methods for specific subclasses.

Asymptotic complexity of the proposed decoder is better than recent algorithms.

A new Plotkin-like construction for matrix rank metric codes is proposed with an efficient decoder.

Abstract

In 2021, Augot, Couvreur, Lavauzelle and Neri introduced a new class of rank metric codes which can be regarded as rank metric counterparts of Reed-Muller codes. Given a finite Galois extension $\mathbb{L} / \mathbb{K}$, these codes are defined as some specific $\mathbb{L}$-subspaces of the twisted group algebra $\mathbb{L} [\textrm{G}]$. We investigate the decoding of such codes in the "binary" case, \emph{i.e.,} when $\textrm{G} = (\mathbb{Z}/2\mathbb{Z})^m$. Our approach takes its inspiration from the decoding of Hamming metric binary Reed-Muller codes using their recursive Plotkin "$(u ~|~ u+v)$" structure. If our recursive algorithm restricts to a specific subclass of rank metric Reed-Muller codes, its asymptotic complexity beats that of the recently proposed decoding algorithm for arbitrary rank metric Reed-Muller codes based on Dickson matrices. Also, this decoder is of completely different nature and leads a natural rank metric counterpart of the Plotkin construction. To illustrate this, we also propose a generic Plotkin-like construction for matrix rank metric codes with an associate decoder, which can be applied to any pair of codes equipped with an efficient decoder.