An improvement to the Haroutunian bound for anytime coding systems

In the study of error exponents, the Haroutunian exponent is encountered as an upper bound for several point-to-point communication problems over DMCs including block coding with feedback and fixed-delay coding. For symmetric channels, such as the BSC or BEC, the Haroutunian exponent is equal to the sphere-packing exponent. But for asymmetric channels, such as the Z-channel, the Haroutunian bound is strictly larger than the sphere-packing exponent. It is generally believed that the sphere-packing bound should hold for these problems, even though they are different from the problem of block coding without feedback. The fundamental difficulty in these problems is that the distribution of the input is not known during the error event, and unlike symmetric channels, there is no `universally good´ input distribution like the uniform distribution. The result is that a worst-case assumption is made on the input distribution to give the Haroutunian bound, even though the resulting input distribution is useless for communication purposes. In order to make progress on this issue, we study an extended notion of fixed-delay codes called anytime codes, a class of codes that indirectly enforce the property that nontrivial communication is attempted during the error event. For this class of codes, we give a new upper bound to the error exponent that strictly improves on the Haroutunian bound for asymmetric channels. While the new exponent still does not reach sphere-packing, we show that the ratio of the two exponents approaches 1 as the rate approaches capacity for Z-channels. This fact may have an interesting consequence for the viewpoint of maximum achievable rate for a given delay and desired error probability. Additionally, the improved exponent yields a tighter bound for a notion of sufficiency of a channel for control purposes.