Finite-Blocklength and Error-Exponent Analyses for LDPC Codes in Point-to-Point and Multiple Access Communication

Abstract

This paper applies error-exponent and dispersionstyle analyses to derive finite-blocklength achievability bounds for low-density parity-check (LDPC) codes over the point-to-point channel (PPC) and multiple access channel (MAC). The errorexponent analysis applies Gallager's error exponent to bound achievable symmetrical and asymmetrical rates in the MAC. The dispersion-style analysis begins with a generalization of the random coding union (RCU) bound from random code ensembles with i.i.d. codewords to random code ensembles in which codewords may be statistically dependent; this generalization is useful since the codewords of random linear codes such as random LDPC codes are dependent. Application of the RCU bound yields improved finite-blocklength error bounds and asymptotic achievability results for i.i.d. random codes and new finiteblocklength error bounds and achievability results for LDPC codes. For discrete, memoryless channels, these results show that LDPC codes achieve first-and second-order performance that is optimal for the PPC and identical to the best-prior results for the MAC.