  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 to provide support for computations involving more than six
transmissions.
  We present two versions of upper and lower bounds for the two-transmission
case. One of the versions is extended to the general case of $m$ transmissions
where $m \geq 1$. Computing the general bounds requires minimization of
exponential functions with the auxiliary parameters, but is less complex and
more stable than multiple rounds of numerical integration. These bounds also
provide a good estimate of the expected throughput and expected latency, which
are useful for optimization purposes.
