11.39. 0; not a field.

11.43. m = 2, 4, pr or 2pr , where p is an odd prime; see Weiss [12, Th. 4–6–10].

11.47. α = √2 + √−3.

11.49. All elements of GF (32) except 0 and 1 are primitive.

11.65.The output has cycle length 7 and repeats the sequence 1101001, starting at the right.

12.3.Use GF(8) = {0, 1, α, 1 + α, α2, 1 + α2, α + α2, 1 + α + α2} where α3 = α + 1.

A, B, C, and D are the four brands of cereal.

A, B, C, and D are the four different types of music, and 0 refers to

no music.

12.15. y = αx. 12.17. y = (α + 2)x + 2α. 12.21. 1.

CHAPTER 14

14.1. 010, 001, 111.

14.3.The checking is done modulo 9, using the fact that any integer is congruent to the sum of its digits modulo 9.

14.5.(1 2 3 4 5 6 7 8 9 10) modulo 11. It will detect one error but not correct any.

14.7. 000000, 110001, 111010, 001011, 101100, 011101, 010110, 100111.

14.9.101, 001, 100.

14.11.Minimum distance = 3. It detects two errors and corrects one error.

14.23.(a) 56; (b) 7; (c) 27 = 128, (d) 8/9; (e) it will detect single, double, triple, and any odd number of errors; (f) 1.

14.25. No.

14.29. x8 + x4 + x2 + x + 1 and x10 + x9 + x8 + x6 + x5 + x2 + 1.

14.31. x5 + x4 + x3 + x2 + 1.