 32, t = 3, and let y by a root of the primitive polynomial z5++22+1 over GF(2). Then yn can be generated by a Galois-configured, binary LFSR having characteris- tic polynomial z5 + z2 + 1, as noted in Section 5.5. Hence, let yn=y's (5.369) where Sn is a state vector over GF(2) representing the contents of the five stages of the LFSR at time n. The companion matrix for this register is (see (5.36)) 00001 10000 A=01001 (5.370a) 00100 00010 and from the development of Section 5.8.2, Sn+1 - As, (5.370b) Squaring yn as represented in (5.369) and using the fact that 25=22+1 to reduce the resultant powers of y, gives S2n = Ss (5.371) where 10001 000010 S=01010 (5.372) 00011 00100 Let yn and ki be coefficient vectors for the representations (5.373) Then it is easily verified that A block diagram of the generator in matrix-vector terms is shown in Figure 5.19. The complexity of this mechanization comes from the require- ment that the value of k2 be changeable, as part of the sequence selection process. Were it not for this specification, the quantity in brackets in (5.374) could be reduced to a single binary matrix, making the generator structure quite simple. FREQUENCY SYNTHESIZER (one-to-one mapping) n + + VECTOR ADDER SEQUENCE SELECTOR ( k0,"2) S n+ T Random Initializer T Shift Register Figure 5.19. Matrix/vector description of a Reed-Solomon FH SSMA PN generator for Example 5.17.All arithmetic operations are in GF(2). Pseudonoise Generators There is a simple tradeoff in the design shown in Figure 5.19, involving generator complexity and the size of the sequence set. If the switches k20, k24 are left permanently open, thereby eliminating the need for five matrix multipliers, then the input to the vector adder is ko + S, the thirty- two possible choices of ko corresponding to the thirty-two period-31 sequences of a t = 2 Reed-Solomon design with Hmin = 30. If instead ko and only the k21 switch are enabled, then the matrix multiply S is required, and there are sixty-four possible sequences that can be generated, a subse