Iterative List-Decoding of Gabidulin Codes via Gr\"obner Based Interpolation

TL;DR

This paper presents an iterative list-decoding algorithm for Gabidulin codes using Gr"obner basis interpolation, enabling efficient decoding by processing received data step-by-step to find all closest codewords under the rank metric.

Contribution

It introduces a novel iterative parametrization method leveraging Gr"obner basis interpolation for list decoding Gabidulin codes, improving decoding efficiency.

Findings

Decodes Gabidulin codes iteratively with polynomial construction

Produces a list of all closest codewords under rank metric

Demonstrates effectiveness of Gr"obner basis approach in decoding

Abstract

We show how Gabidulin codes can be list decoded by using an iterative parametrization approach. For a given received word, our decoding algorithm processes its entries one by one, constructing four polynomials at each step. This then yields a parametrization of interpolating solutions for the data so far. From the final result a list of all codewords that are closest to the received word with respect to the rank metric is obtained.