Chernoff bounds for analysis of rate-compatible sphere-packing with numerous transmissions

Recent results by Chen et al. and Polyanskiy et al. explore using feedback to approach capacity with short blocklengths. This paper explores Chernoff bounding techniques to extend the rate-compatible sphere-packing (RCSP) analysis proposed by Chen et al. to scenarios involving numerous retransmissions and different step sizes in each incremental retransmission. Williamson et al. employ exact RCSP computations for up to six transmissions. However, exact RCSP computation with more than six retransmissions becomes unwieldy because of joint error probabilities involving numerous chi-squared distributions. This paper explores Chernoff approaches for upper and lower bounds on the error probability to provide support for computations involving more than six transmissions. We present two versions of upper and lower bounds on the error probability for the two-transmission case. One of the versions is extended to the general case of m transmissions where m ≥ 1. Computing the bounds for general m requires minimization of exponential functions with the auxiliary parameters. The numerical results, however, show that weakening the bounds by considering marginal probabilities and the case of two transmissions is already tight. These bounds also provide good estimates of the expected throughput and expected latency, which are useful for optimization purposes.