or- relation and out-of-phase autocorrelation values taken from the set -1,-1 + - 2(L+2)/2]_ Including the component sequences, Gold's design contains 24 + 1 sequences, a sufficient number for Sidelnikov's bound (5.245) to apply. Specializing the bound to this case gives (5.288) where Pmax is defined in (5.232)-(5.235). For L 3, this implies that Pmax is greater than -1 + 2(L+1)/2 and, since the binary sequence set's period is odd, Pmax must be odd, yielding (5.289) Gold sequences with L odd have Pmax satisfying this bound with equality, and, hence, are an optimal design from the viewpoint of minimizing the absolute periodic correlation parameter Pmax Theorem 5.20 indicates that only 2L - 2L-e - 1 of the compound sequences in the Gold set are balanced, with the imbalance in the remain- ing 2L-e sequences being on the order of 2(L+2)/21_ Example 5.15 (the small set of Kasami sequences). Kasami [30] in 1966 reported results on the enumeration of weights of linear cyclic codes over GF(2). He noted that this problem is equivalent (as in Theorem 5.20) to the Pseudonoise Generators evaluation of cross-correlation or out-of-phase autocorrelation between two binary (+1) sequences, and produced forms of the results described in Theorems 5.16 and 5.18. The small set of Kasami sequences is a binary linear design whose com- ponent sequences are (1) an m-sequence with period 2L - 1, L even, and characteristic polynomial ma(z), a E GF(2), and (2) an m-sequence with shorter period 2L12 - 1 and characteristic polynomial ma(z) where r=242+1 (5.290) being an element of GF(21/2). The period of the latter m-sequence divides the longer period of the first m-sequence (see Lemma 5.3), and Theorem 5.18 states that the periodic cross-correlation of these two m-sequences, over the longer period, is - -1 2 2 at all relative phase shifts. Application of Theorem 5.19 to these m-sequences, with p = p' = 2, Nm = 2L - 1, and N = 2L12 - 1, yields a binary linear design now referred to as the small set of Kasami sequences, with all cross-correlat