se noise and tone jam- ming. Minima occur at RS code rates k/n > .8, with Mopt >6. and tone jamming. And since the convolutional codes performed generally better than the one class of block codes considered (i.e., the Reed-Solomon codes), we will restrict this exercise to random (n, k) block codes with char- acters over GF(2*). So, for each block of k K-bit information symbols, the encoder generates a block of n K-bit code symbols, according to a randomly generated mapping as in [9, pp. 309-320]; this corresponds to a code rate M-ary use information channel bits (2.178) in the notation of Section 2.3.3. The random coding scheme selects a set of Mk = 2Kk = 2nR codewords {si} E GF(2*), with a mapping of the j-th mes- sage, Mj, into codeword Sj. If all sets of codewords and messages are equally probable, the union bound on the word error rate is [9, (5.47)] Pw = Pr{word error|m}} (l#j) < (2nR'-1)2-nR (2.179) where P2(S1,Sj) is the probability that codeword sj is sent and the decoder Non-Coherent Frequency-Hopped Systems decides message m was intended, the expectation is over the codeword set {si}, and R is the M-ary channel cutoff rate in bits/channel use (see Chapter 4, Part 1). We can convert R to the parameter D used in this chapter accord- ing to (see (4.27), Part 1) (2.180) which implies that (2.181) By analogy with (2.20a), with Mk = 2nR replacing M, (2.182) Suppose we concatenate the (n,k) random block code with an m-diver- sity inner code. That is, the block code operates over an M-ary channel with D replaced by Dm as in Section 2.3.3. When this diversity is optimized, (2.162)-(2.164) apply for worst case noise and tone jamming: substituting these expressions into (2.182), we find that (2.183) As an example, (2.183) was used to plot an uppebound on Eb/N1 required to achieve Pb = 10-5 as a function of R' with parameter n, for K = log2M = 2 and worst case partial-band noise jamming. Figure 2.104 shows that for each n, the required Eb/N, upperbound is minimized at an interior code rate R' = Ropt If we compute M