In a remarkable paper published in 1976, Burnashev determined the reliability function of variable-length block codes over discrete memoryless channels (DMCs) with feedback. Subsequently, an alternativeachievabilityproof was obtained by Yamamoto and Itoh via a particularly simple and instructive scheme. Their idea is to alternate between a communication and a confirmation phase until the receiver detects the codeword used by the sender to acknowledge that the message is correct. We provide aconversethat parallels the Yamamoto-Itoh achievability construction. Besides being simpler than the original, the proposed converse suggests that a communication and a confirmation phase are implicit in any scheme for which the probability of error decreases with the largest possible exponent. The proposed converse also makes it intuitively clear why the terms that appear in Burnashev's exponent are necessary.

Feedback coupled with variable-length codes can substantially increase the reliability of a discrete memoryless channel (DMC). Burnashev, in a remarkable paper published in 1976, derived an asymptotically achievable lower bound to the average blocklength needed for a system that communicates at a specified rate and achieves a given error probability. We offer an alternative proof of the lower bound. Our proof is simpler than the original, and clarifies the roles of the quantites that appear in the bound by relating one to uncertainty reduction and the other to binary hypothesis testing. In addition, our derivation of the lower bound closely parallels a derivation of an upper bound by Yamamoto and Itoh.

In this work, we study coding over a class of two-user broadcast channels (BCs) with additive white Gaussian noise and multiplicative fading known at the receivers only. Joint decoding of low-density parity-check (LDPC) codes is analyzed. The message update rule at the mapping node linking the users' codes is derived and is found to exhibit an interesting soft interference cancellation property. High performance codes are found using the differential evolution optimization technique and extrinsic information transfer analysis adapted to our multiuser setting. The optimized codes have rates very close to the boundary of the achievable region for binary constrained input for both faded and unfaded channels. Simulation results for moderate block lengths show that our codes operate within less than 1 dB of their respective threshold.