Decoder Ties Do Not Affect the Error Exponent of the Memoryless Binary Symmetric Channel

TL;DR

This paper proves that decoder ties do not influence the error exponent in the memoryless binary symmetric channel, showing that ties only affect subexponential error behavior.

Contribution

It establishes that for BSC, decoder ties do not impact the exponential decay rate of error probability, extending the generalized Poor-Verdu bound.

Findings

Decoder ties do not affect the error exponent in BSC.

The error probability deviation from the bound grows at most linearly with blocklength.

Ties influence only subexponential error behavior, not the exponential decay rate.

Abstract

The generalized Poor-Verdu error lower bound established in [1] for multihypothesis testing is studied in the classical channel coding context. It is proved that for any sequence of block codes sent over the memoryless binary symmetric channel (BSC), the minimum probability of error (under maximum likelihood decoding) has a relative deviation from the generalized bound that grows at most linearly in blocklength. This result directly implies that for arbitrary codes used over the BSC, decoder ties can only affect the subexponential behavior of the minimum probability of error.