Parallel Concatenation of Non-Binary Linear Random Fountain Codes with Maximum Distance Separable Codes

TL;DR

This paper introduces a parallel concatenation coding scheme combining MDS and linear random fountain codes, significantly reducing decoding failure probability and approaching ideal fountain code performance, with an efficient decoding algorithm for Reed-Solomon codes.

Contribution

It proposes a novel parallel concatenation of MDS and linear random fountain codes, providing bounds on failure probability and demonstrating substantial performance improvements.

Findings

Decoding failure probability is reduced by four orders of magnitude compared to standard fountain codes.

Performance approaches that of ideal fountain codes at higher field orders and moderate erasures.

An efficient decoding algorithm is developed for Reed-Solomon codes.

Abstract

The performance and the decoding complexity of a novel coding scheme based on the concatenation of maximum distance separable (MDS) codes and linear random fountain codes are investigated. Differently from Raptor codes (which are based on a serial concatenation of a high-rate outer block code and an inner Luby-transform code), the proposed coding scheme can be seen as a parallel concatenation of a MDS code and a linear random fountain code, both operating on the same finite field. Upper and lower bounds on the decoding failure probability under maximum-likelihood (ML) decoding are developed. It is shown how, for example, the concatenation of a $(15,10)$ Reed-Solomon (RS) code and a linear random fountain code over a finite field of order $16$, $\mathbb {F}_{16}$, brings to a decoding failure probability $4$ orders of magnitude lower than the one of a linear random fountain code for the same receiver overhead in a channel with a erasure probability of $\epsilon=5\cdot10^{-2}$. It is illustrated how the performance of the novel scheme approaches that of an idealized fountain code for higher-order fields and moderate erasure probabilities. An efficient decoding algorithm is developed for the case of a (generalized) RS code.