Channel Coding Rate in the Finite Blocklength Regime

This paper investigates the maximal channel coding rate achievable at a given blocklength and error probability. For general classes of channels new achievability and converse bounds are given, which are tighter than existing bounds for wide ranges of parameters of interest, and lead to tight approximations of the maximal achievable rate for blocklengths n as short as 100. It is also shown analytically that the maximal rate achievable with error probability ¿ isclosely approximated by C - ¿(V/n) Q^-1(¿) where C is the capacity, V is a characteristic of the channel referred to as channel dispersion , and Q is the complementary Gaussian cumulative distribution function.

I. Introduction

The proof of the channel coding theorem involves three stages: •

Converse: an upper bound on the size of any code with given arbitrary blocklength and error probability.

•

Achievability: a lower bound on the size of a code that can be guaranteed to exist with given arbitrary blocklength and error probability.

•

Asymptotics: the bounds on the log size of the code normalized by blocklength asymptotically coincide as a result of the law of large numbers (memoryless channels) or another ergodic theorem (for channels with memory).