Composition bounds for channel block codes

Abstract

The performance of Channel block codes for a general channel is studied by examining the relationship between the rate of a code, the joint composition of pairs of codewords, and the probability of decoding error. At fixed rate, lower bounds and upper bounds, both on minimum Bhattacharyya distance between codewords and on minimum equivocation distance between codewords, are derived. These bounds resemble, respectively, the Gilbert and the Elias bounds on the minimum Hamming distance between codewords. For a certain large class of channels, a lower bound on probability of decoding error for low-rate channel codes is derived as a consequence of the upper bound on Bhattacharyya distance. This bound is always asymptotically tight at zero rate. Further, for some channels, it is asymptotically tighter than the straight line bound at low rates. Also studied is the relationship between the bounds on codeword composition for arbitrary alphabets and the expurgated bound for arbitrary channels having zero error capacity equal to zero. In particular, it is shown that the expurgated reliability-rate function for blocks of letters is achieved by a product distribution whenever it is achieved by a block probability distribution with strictly positive components.