Bounds on the Error Probability of Raptor Codes under Maximum Likelihood Decoding

TL;DR

This paper derives tight upper and lower bounds on the decoding failure probability for various nonbinary Raptor code constructions under maximum likelihood decoding, aiding in their design.

Contribution

It provides the first comprehensive bounds for multiple Raptor code variants, including the state-of-the-art RaptorQ, under maximum likelihood decoding.

Findings

Bounds are tight and accurate.

Different Raptor code constructions are analyzed.

Results assist in optimizing Raptor code design.

Abstract

In this paper upper and lower bounds on the probability of decoding failure under maximum likelihood decoding are derived for different (nonbinary) Raptor code constructions. In particular four different constructions are considered; (i) the standard Raptor code construction, (ii) a multi-edge type construction, (iii) a construction where the Raptor code is nonbinary but the generator matrix of the LT code has only binary entries, (iv) a combination of (ii) and (iii). The latter construction resembles the one employed by RaptorQ codes, which at the time of writing this article represents the state of the art in fountain codes. The bounds are shown to be tight, and provide an important aid for the design of Raptor codes.