# A Lower Bound on the Error Exponent of Linear Block Codes over the Erasure Channel

## TL;DR

This paper derives a lower bound on the error exponent for linear block codes over erasure channels, providing insights into decoding thresholds and analyzing specific code ensembles like Raptor codes.

## Contribution

It introduces a new lower bound on the ML decoding error exponent for linear block codes over erasure channels, applicable to various code ensembles.

## Key findings

- Lower bound is positive within certain erasure probability intervals.
- Analytical solution for linear random parity-check codes.
- Lower bound on Raptor codes' ML decoding threshold via nonlinear equations.

## Abstract

A lower bound on the maximum likelihood (ML) decoding error exponent of linear block code ensembles, on the erasure channel, is developed. The lower bound turns to be positive, over an ensemble specific interval of erasure probabilities, when the ensemble weight spectral shape function tends to a negative value as the fractional codeword weight tends to zero. For these ensembles we can therefore lower bound the block-wise ML decoding threshold. Two examples are presented, namely, linear random parity-check codes and fixed-rate Raptor codes with linear random precoders. While for the former a full analytical solution is possible, for the latter we can lower bound the ML decoding threshold on the erasure channel by simply solving a 2 x 2 system of nonlinear equations.