Decoding High-Order Interleaved Rank-Metric Codes

TL;DR

This paper introduces a linear-algebraic decoding algorithm for high-order interleaved rank-metric codes that corrects errors up to a certain weight, applicable to any rank-metric code, not just Gabidulin codes.

Contribution

It adapts a Hamming-metric decoder to rank metric, providing a universal, explicit, and efficient decoding method for interleaved codes of high order.

Findings

Decodes all rank errors up to $d-2$ in homogeneous interleaved codes.

Works for any rank-metric code, not only Gabidulin codes.

Provides a lower bound on decoding success probability for random errors.

Abstract

This paper presents an algorithm for decoding homogeneous interleaved codes of high interleaving order in the rank metric. The new decoder is an adaption of the Hamming-metric decoder by Metzner and Kapturowski (1990) and guarantees to correct all rank errors of weight up to $d-2$ whose rank over the large base field of the code equals the number of errors, where $d$ is the minimum rank distance of the underlying code. In contrast to previously-known decoding algorithms, the new decoder works for any rank-metric code, not only Gabidulin codes. It is purely based on linear-algebraic computations, and has an explicit and easy-to-handle success condition. Furthermore, a lower bound on the decoding success probability for random errors of a given weight is derived. The relation of the new algorithm to existing interleaved decoders in the special case of Gabidulin codes is given.