Recursive decoding of binary rank Reed-Muller codes and Plotkin construction for matrix codes
Alain Couvreur, Rakhi Pratihar
TL;DR
The paper addresses decoding of binary-like rank-metric Reed–Muller codes defined via skew group algebras and introduces a recursive Plotkin-inspired decoding that corrects up to half the minimum distance with improved asymptotic complexity over prior Dickson-matrix approaches. The method relies on a recursive Kummer/Artin–Schreier structure to fold and erasure-decode, yielding a rank-metric analogue of the Plotkin (u|u+v) construction that extends to matrix codes. It provides detailed complexity analyses, fast-syndrome techniques, and probabilistic guarantees for folding-preservation of rank, along with experimental validation. The Plotkin construction is then adapted to matrix codes over finite fields, producing new families of efficiently decodable rank-metric codes that are not equivalent to Gabidulin codes, with practical decoding strategies and implications for code design in rank-metric settings.
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.