Row Reduction Applied to Decoding of Rank Metric and Subspace Codes

TL;DR

This paper introduces a novel row reduction method for skew polynomial matrices to improve decoding algorithms for rank metric and subspace codes, achieving efficient complexity in decoding and interpolation tasks.

Contribution

It develops a flexible row reduction approach for skew polynomial matrices and applies it to enhance decoding algorithms for interleaved Gabidulin and Mahdavifar--Vardy codes.

Findings

Decoding of interleaved Gabidulin codes achieved with $O( ext{input size}^2)$ complexity.

List-decoding of Mahdavifar--Vardy codes performed with $O( ext{constraints} imes n^2)$ complexity.

New algorithms improve efficiency of decoding and interpolation in rank metric and subspace coding.

Abstract

We show that decoding of $\ell$-Interleaved Gabidulin codes, as well as list-$\ell$ decoding of Mahdavifar--Vardy codes can be performed by row reducing skew polynomial matrices. Inspired by row reduction of $\F[x]$ matrices, we develop a general and flexible approach of transforming matrices over skew polynomial rings into a certain reduced form. We apply this to solve generalised shift register problems over skew polynomial rings which occur in decoding $\ell$-Interleaved Gabidulin codes. We obtain an algorithm with complexity $O(\ell \mu^2)$ where $\mu$ measures the size of the input problem and is proportional to the code length $n$ in the case of decoding. Further, we show how to perform the interpolation step of list-$\ell$-decoding Mahdavifar--Vardy codes in complexity $O(\ell n^2)$, where $n$ is the number of interpolation constraints.