te the vector by e. Based on the left side of (5.363), (5.364) The Hamming distance over one period between {f(})} and {f(d)} at rela- tive shift T is given by (5.365) Notice that Pe(s") )=Pa(2") Pe-d()= 2" is a root of Pe-d(z). (5.366) The number of roots in GF(q) of such a polynomial is bounded by t - 1, its maximum possible degree. Use of this fact in (5.365) leads to the distance bound for Reed-Solomon sequences, namely Hmin = 9 - t. (5.367) Since the sequences' alphabet size is q and their period is q - 1, the bound on Hmin based on Theorem 5.24 is min 9q - 2 (5.368) In view of (5.367) and (5.368), the Reed-Solomon design with t = 2 produces a set of q sequences over GF(q) with Hmin = q - 2, indicating that in a com- parison of two sequences, at any relative shift T, the t=2 design guarantees at most one frequency match. Frequency-Hopping Multiple-Access Designs conceptual block diagram of a t = 3 Reed-Solomon sequence genera- tor is shown in Figure 5.18. In this form, a squaring device is required to ran- domly initialize this generator within a particular sequence. Another configuration can be obtained by combining the two single-stage q-ary LFSRs into a single two-stage LFSR. Still another possibility is to square the contents yn of the first single-stage q-ary register to directly determine the input to the C2 multiplier, thereby eliminating the second register and the need for a squarer during initialization. Notice that the generator for a t = 2 design is contained within that shown in Figure 5.18, and is determined by FREQUENCY SYNTHESIZER (one-to-one mapping) palyntty SEQUENCE SELECTOR (CO,C2) squarer RANDOM INITIALIZER T 2(n+T) Y n-T Figure 5.18. Mathematical block diagram of a Reed-Solomon FH PN generator. All arithmetic operations are in GF(q). Pseudonoise Generators setting C2 to zero permanently, thereby eliminating the need for the right- hand (22) shift register. Example 5.17. A Reed-Solomon design with q being a power of 2 results in a relatively simple implementation. In this example let q =