Abstract Algebra Theory and Applications - Computer Science ...

362 CHAPTER 20 FINITE FIELDS22. Show that every element in GF(p n ) can be written in the form a p for someunique a ∈ GF(p n ).23. Let E and F be subfields of GF(p n ). If |E| = p r and |F | = p s , what is theorder of E ∩ F ?24. Wilson’s Theorem. Let p be prime. Prove that (p − 1)! ≡ −1 (mod p).25. If g(t) is the minimal generator polynomial for a cyclic code C in R n , provethat the constant term of g(x) is 1.26. Often it is conceivable that a burst of errors might occur during transmission,as in the case of a power surge. Such a momentary burst of interferencemight alter several consecutive bits in a codeword. Cyclic codes permit thedetection of such error bursts. Let C be an (n, k)-cyclic code. Prove thatany error burst up to n − k digits can be detected.27. Prove that the rings R n and Z n 2 are isomorphic as vector spaces.28. Let C be a code in R n that is generated by g(t). If 〈f(t)〉 is another code inR n , show that 〈g(t)〉 ⊂ 〈f(t)〉 if and only if f(x) divides g(x) in Z 2 [x].29. Let C = 〈g(t)〉 be a cyclic code in R n and suppose that x n − 1 = g(x)h(x),where g(x) = g 0 + g 1 x + · · · + g n−k x n−k and h(x) = h 0 + h 1 x + · · · + h k x k .Define G to be the n × k matrix⎛G =⎜⎝and H to be the (n − k) × n matrixH =⎛⎜⎝⎞g 0 0 · · · 0g 1 g 0 · · · 0.. . .. .g n−k g n−k−1 · · · g 00 g n−k · · · g 1. ⎟. . .. . ⎠0 0 · · · g n−k0 · · · 0 0 h k · · · h 00 · · · 0 h k · · · h 0 0· · · · · · · · · · · · · · · · · · · · ·h k · · · h 0 0 0 · · · 0(a) Prove that G is a generator matrix for C.(b) Prove that H is a parity-check matrix for C.(c) Show that HG = 0.⎞⎟⎠ .