How to beat the sphere-packing bound with feedback

The sphere-packing bound $E_{sp}(R)$ bounds the reliability function for fixed-length block-codes. For symmetric channels, it remains a valid bound even when strictly causal noiseless feedback is allowed from the decoder to the encoder. To beat the bound, the problem must be changed. While it has long been known that variable-length block codes can do better when trading-off error probability with expected block-length, this correspondence shows that the {\em fixed-delay} setting also presents such an opportunity for generic channels. While $E_{sp}(R)$ continues to bound the tradeoff between bit error and fixed end-to-end latency for symmetric channels used {\em without} feedback, a new bound called the ``focusing bound'' gives the limits on what can be done with feedback. If low-rate reliable flow-control is free (ie. the noisy channel has strictly positive zero-error capacity), then the focusing bound can be asymptotically achieved. Even when the channel has no zero-error capacity, it is possible to substantially beat the sphere-packing bound by synthesizing an appropriately reliable channel to carry the flow-control information.