Asymptotic expansion and error exponent for two-phase feedback codes on DMCs

This paper studies variable-length coding with noise-less feedback for discrete memoryless channels. Yamamoto and Itoh´s two-phase scheme achieves the optimal error-exponent, but due to the block-coding nature it is not optimal in the expansion of the message size logM. Polyanskiy et al. showed that with feedback, the back-off from capacity is logarithmic in the expected latency ℓ. The O(log ℓ) back-off is achieved by using an incremental redundancy (IR) scheme that only utilizes feedback to determine the stopping time. However, the achievable error-exponent of the IR scheme is not optimal. This paper shows that a two-phase coding scheme where each phase uses an IR scheme achieves the optimal error-exponent while maintaining an expansion on the message size that yields the O(log ℓ) back-off.