List and Unique Error-Erasure Decoding of Interleaved Gabidulin Codes with Interpolation Techniques

TL;DR

This paper introduces a novel interpolation-based decoding method for interleaved Gabidulin codes, enabling efficient list and probabilistic unique decoding, and extends it to handle errors and erasures with bounded failure probability.

Contribution

The paper presents a new interpolation-based decoding framework for interleaved Gabidulin codes, including list and probabilistic unique decoding, with extensions for erasures and error correction.

Findings

Decoding reduces to solving linear systems via interpolation.

The approach enables list decoding with potentially unbounded list size.

An upper bound on failure probability for the unique decoder is provided.

Abstract

A new interpolation-based decoding principle for interleaved Gabidulin codes is presented. The approach consists of two steps: First, a multi-variate linearized polynomial is constructed which interpolates the coefficients of the received word and second, the roots of this polynomial have to be found. Due to the specific structure of the interpolation polynomial, both steps (interpolation and root-finding) can be accomplished by solving a linear system of equations. This decoding principle can be applied as a list decoding algorithm (where the list size is not necessarily bounded polynomially) as well as an efficient probabilistic unique decoding algorithm. For the unique decoder, we show a connection to known unique decoding approaches and give an upper bound on the failure probability. Finally, we generalize our approach to incorporate not only errors, but also row and column erasures.