List Decoding of Matrix-Product Codes from nested codes: an application to Quasi-Cyclic codes

TL;DR

This paper presents a list decoding algorithm for matrix-product codes constructed from nested codes, including quasi-cyclic codes, with analysis of decoding performance and extensions to unique decoding.

Contribution

It introduces a novel list decoding algorithm for matrix-product codes from nested codes and extends it to quasi-cyclic codes with polynomial units.

Findings

Probability estimates for multiple codeword outputs with Reed-Solomon codes

Extension of decoding algorithm to quasi-cyclic codes with polynomial units

Implementation of unique decoding for matrix-product codes with polynomial units

Abstract

A list decoding algorithm for matrix-product codes is provided when $C_1,..., C_s$ are nested linear codes and $A$ is a non-singular by columns matrix. We estimate the probability of getting more than one codeword as output when the constituent codes are Reed-Solomon codes. We extend this list decoding algorithm for matrix-product codes with polynomial units, which are quasi-cyclic codes. Furthermore, it allows us to consider unique decoding for matrix-product codes with polynomial units.