On the Parity-Check Density and Achievable Rates of LDPC Codes for Memoryless Binary-Input Output-Symmetric Channels

We introduce upper bounds on the achievable rates of binary linear block codes under maximum-likelihood (ML) decoding, and lower bounds on the asymptotic density of their parity-check matrices; the transmission of these codes is assumed to take place over an arbitrary memoryless binary-input output-symmetric channel. The bounds hold for every sequence of binary linear block codes. The new bounds tighten previously reported results, and enable to obtain tighter information-theoretic bounds on the thresholds of sequences of binary linear block codes under ML decoding. The bounds are applied to low-density parity-check codes, and their improvement is exemplified numerically.