Distance Spectrum of Fixed-Rate Raptor Codes with Linear Random Precoders

TL;DR

This paper derives the distance spectrum and asymptotic weight distribution of fixed-rate Raptor codes with linear random precoders, providing conditions for positive minimum distance and linear growth with block length.

Contribution

It introduces a new analytical framework for the average distance spectrum and minimum distance properties of fixed-rate Raptor codes with linear random outer codes.

Findings

Derived an expression for the average distance spectrum.

Established a necessary and sufficient condition for positive typical minimum distance.

Showed that minimum distance grows linearly with block length under certain conditions.

Abstract

Raptor code ensembles with linear random outer codes in a fixed-rate setting are considered. An expression for the average distance spectrum is derived and this expression is used to obtain the asymptotic exponent of the weight distribution. The asymptotic growth rate analysis is then exploited to develop a necessary and sufficient condition under which the fixed-rate Raptor code ensemble exhibits a strictly positive typical minimum distance. The condition involves the rate of the outer code, the rate of the inner fixed-rate Luby Transform (LT) code and the LT code degree distribution. Additionally, it is shown that for ensembles fulfilling this condition, the minimum distance of a code randomly drawn from the ensemble has a linear growth with the block length. The analytical results can be used to make accurate predictions of the performance of finite length Raptor codes. These results are particularly useful for fixed-rate Raptor codes under maximum likelihood erasure decoding, whose performance is driven by their weight distribution.