Fast Decoding of Codes in the Rank, Subspace, and Sum-Rank Metric

TL;DR

This paper introduces faster decoding algorithms for codes in rank, subspace, and sum-rank metrics by reducing core computational problems to approximant bases over skew polynomial rings, leveraging a new skew PM-Basis algorithm.

Contribution

It presents a novel skew PM-Basis algorithm and unified approach to speed up decoding algorithms across multiple code classes using skew polynomial computations.

Findings

Decoding algorithms are significantly faster due to new approximant basis computations.

The skew PM-Basis algorithm generalizes existing polynomial algorithms to skew polynomials.

The methods have broader applications in code decoding and skew polynomial evaluation.

Abstract

We speed up existing decoding algorithms for three code classes in different metrics: interleaved Gabidulin codes in the rank metric, lifted interleaved Gabidulin codes in the subspace metric, and linearized Reed-Solomon codes in the sum-rank metric. The speed-ups are achieved by new algorithms that reduce the cores of the underlying computational problems of the decoders to one common tool: computing left and right approximant bases of matrices over skew polynomial rings. To accomplish this, we describe a skew-analogue of the existing PM-Basis algorithm for matrices over ordinary polynomials. This captures the bulk of the work in multiplication of skew polynomials, and the complexity benefit comes from existing algorithms performing this faster than in classical quadratic complexity. The new algorithms for the various decoding-related computational problems are interesting in their own and have further applications, in particular parts of decoders of several other codes and foundational problems related to the remainder-evaluation of skew polynomials.