 can be derived as follows: w # v (5.359) Both h( , , ) and h'( , ) are zero if their two arguments (scalars and vec- tors respectively) are equal, and are one otherwise. At (1) in (5.359), the Hamming distance-preserving property of one-to-one mappings is used; (2) is based on the shift-and-add property (5.63) of m-sequences; and (3) results from the balance property (5.61) of m-sequences. The minimum Hamming distance Hmin for this design is q1 - 1 - qL-1, indi- cating that at most qL-1 matching frequencies appear in a comparison of two sequences over one period, at any relative shift. The quantities B and a of (5.353) are q' - 1 and qL-J - 1 respectively, and, hence, using the fact that HA and H must be integer, it can be verified that these Hamming measures for this design satisfy the bounds of Theorem 5.4 and Theorem 5.24 with equality. 5.8.4 Reed-Solomon Sequences This algebraic design is based on mechanizing calculations in GF(q), and employing a one-to-one mapping s(.) from elements of GF(q) to a set of output frequency values. Let y denote a primitive element of GF(q), and define the polynomial Pc(z) of degree t - 1,t>1, over GF(q) to be (5.360) Elements of the sequences in this set are defined by (5.361) for all n, where C is any coefficient vector over GF(q) with C1 = 1. Hence, there are qt-1 sequences in this set. Each of these sequences has period q - 1, the multiplicative order of y. Pseudonoise Generators Two sequences, {f(c)} and {f(d)} are cyclically distinct if (5.362) for some value of n at each possible value of T. This condition (5.362) reduces by (5.360) to (5.363) Since the t sequences {yni},i = 0, 1,...,t - 1, are linearly independent, the condition (5.363) can be satisfied if and only if ciyTi # d for some i at each value of T. Fixing C1 = d1 = 1 for all sequences guarantees that this inequal- ity is satisfied for i = 1 at all T # 0 mod q - 1. When T = 0 mod q - 1, the condition (5.363) reduces to C # d. Hence, the set of sequences with C1 = are cyclically distinct. Let ei = CiYTi and deno