Question: Problem 6. (20 points) Let ac denote a primitive element of GF (22), and let [3 = as, where s : 2' + 1. a.

Problem 6. (20 points) Let ac denote a primitive element of GF (22\"), and let [3 = as, where s : 2'" + 1. a. Note that GF(2m) is a subled of GF(22m), consisting of those elements 1/ E (313027") that sat isfy 1/ m = 1/. Show that )3 E GF(2'"), and that {3 is a primitive element in this eld. b. Consider the binary cyclic code (C of length n = 22'" 1 generated by g(x) = M(x)M5(x), where M\"(x) and Mg(x) are the minimal polynomials of or and B over GF(2). What is the di- mension of this code? c. Now consider the code C\" whose parity-check polynomial is h(x) = Ma(x)Mg (1?). What are the zeros of (3* ? What is the BCH lower bound on the minimum distance of this code

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