Extremes of Error Exponents
This paper determines the range of feasible values of standard error exponents for binary-input memory-less symmetric channels of fixed capacity and shows that extremes are attained by the binary symmetric and the binary erasure channel. The proof technique also provides analogous extremes for other quantities related to Gallager's function, such as the cutoff rate, the Bhattacharyya parameter, and the channel dispersion. In the context of coded communication, the channel coding theorem relates the error probability and the code rate, showing that there exist codes whose error probability tends to zero provided that the code rate is smaller than the channel capacity. For uncoded systems, the error probability and the channel capacity are also related.