akes advantage of the fact that GF(8) is a three- dimensional vector space over GF(2) with basis 1, a9, 18. That is, an element from GF(8) can be written as (5.210) Non-Linear Feed-Forward Logic Table 5.12 A representation for GF(8) elements, and the ROM mapping for Example 5.12. Tr2(23) with Yo, Y1, Y2 in GF(2). Table 5.12 lists this representation along with the ROM mapping, Tr2(23) The multiplications required in the generator of Figure 5.13 are easily mechanized in this representation. (5.211) Similarly, (5.212) This results in the mechanization of Figure 5.14(a). One period of each of the sequences produced in this generator is shown in Figure 5.14(b). Notice the periodically recurring zeros in the GF(8) sequences (T is 9 in this example), this demonstrating the structure exploited in the proof of Theorem 5.12. Tr (an) Figure 5.13. A GMW sequence generator in the Galois configuration with elements in GF(8). Pseudonoise Generators Yo: 011100011001110110000011110010010101001101000010001011011111101 Y1: : 00001111001001010100110100001000101101111110101110001100111011 Y2: : 001110110000011110010010101001101000010001011011111101011100011 Yo: 100111011000001111001001010100110100001000101101111110101110001 f: 00100101010011010000100010110111111010111000110011101100000111 Y2: :000001111001001010100110100001000101101111110101110001100111011 bn: 010011101011101001011100011001111110010010111001110100000001010 Figure 5.14. (a) A GF(2) mechanization of the GMW sequence generator in Example 5.12. (b) Sequences produced in this generator. Since r is 3, the base-2 expansion of r has weight 2, and Theorem 5.13 states that the linear span in this example is 12. The time coefficients c(k, 3) in the powers-of-a expansion of the cubed GF(8) sequence are given in Table 5.13. When the final trace operation to GF(2) is carried out, the remaining cyclic shifts of the binary time coefficients listed in Table 5.12 are added to give twelve distinct coefficients in all. It is obvious from the cyclo- tomic coset representatives in this t