Decoding Interleaved Gabidulin Codes using Alekhnovich's Algorithm

TL;DR

This paper demonstrates how Alekhnovich's algorithm can be applied to decode interleaved Gabidulin codes efficiently, achieving improved complexity over previous methods by leveraging skew polynomial matrix row reduction.

Contribution

It introduces a novel application of Alekhnovich's algorithm for decoding interleaved Gabidulin codes with better asymptotic complexity.

Findings

Decoding algorithm with $O( ext{polylog}(n) imes n^{( ext{omega}+1)/2})$ complexity

Improved exponent in decoding complexity compared to prior work

Proof of applicability of Alekhnovich's algorithm to skew polynomial matrices

Abstract

We prove that Alekhnovich's algorithm can be used for row reduction of skew polynomial matrices. This yields an $O(\ell^3 n^{(\omega+1)/2} \log(n))$ decoding algorithm for $\ell$-Interleaved Gabidulin codes of length $n$, where $\omega$ is the matrix multiplication exponent, improving in the exponent of $n$ compared to previous results.