d (see Appendix 4A) and the union bound which has the gen- eral form (4.25) In addition, in many cases of interest we can reduce the bound based on these two inequalities by a factor of one-half. In all cases where the maxi- mum-likelihood metric is used, the factor of one-half applies [8], [9]. This is shown in Appendix 4B. In all examples given in this book we will introduce this factor of one-half whenever it can be used. 4.3 CUTOFF RATES The parameter D is directly related to the coding channel cutoff rate R0. In general, the channel capacity is the theoretical upper limit on data rates where arbitrarily small bit error probabilities can be achieved with coding [2]. From a practical viewpoint, it is difficult to obtain small bit error prob- abilities with data rates near the channel capacity. Most practical coding tech- niques operate near the smaller data rate R0. Thus, R is known as the practically achievable data rate with coding. In addition, R0 is easier to eval- uate than the capacity of the coding channel. Next we examine the relationship between Ro and the pairwise error bound for the two coded sequences of length N denoted X and x. Assuming the code symbol alphabet consists of M distinct symbols, randomly select the 2N symbols of X and x where each symbol is independently selected with Cutoff Rates uniform probability. The average of the pairwise error bound is then -{1+(1-1) (4.26) where the overbar indicates the average over the coded sequences. The cutoff rate for this case is defined as R0 = log2M - log2{1 + (M - 1)D} bits/symbol (4.27) SO that the average pairwise error probability has the form (4.28) This definition of the cutoff rate is the natural generalization of the usual definition where the maximum-likelihood metric is assumed. Since from (4.20) D is defined for an arbitrary decoding metric, (4.27) likewise has been generalized to apply to arbitrary decoding metrics. For the special case where the maximum-likelihood metric is used and D is given by (4.22), we have the usual definition of