Sub-Quadratic Decoding of Gabidulin Codes

TL;DR

This paper introduces algorithms that enable decoding Gabidulin codes more efficiently by reducing complexity to sub-quadratic time through advanced operations on linearized polynomials.

Contribution

It presents novel fast algorithms for division, evaluation, and interpolation of linearized polynomials, enabling sub-quadratic decoding of Gabidulin codes.

Findings

Decoding complexity reduced to sub-quadratic time

Fast algorithms for linearized polynomial operations

Efficient computation of minimal subspace polynomials

Abstract

This paper shows how to decode errors and erasures with Gabidulin codes in sub-quadratic time in the code length, improving previous algorithms which had at least quadratic complexity. The complexity reduction is achieved by accelerating operations on linearized polynomials. In particular, we present fast algorithms for division, multi-point evaluation and interpolation of linearized polynomials and show how to efficiently compute minimal subspace polynomials.