Let EF256, let be a primitive element of E, and let a = 35. Note that the order of a is 51 (i.e., you are given this fact and do not need to check it or justify it). Let C be the BCH code based on a with designed distance = 5 over F2. In the following, show all your work, especially your orbit calculations. (a) Recall that mi(x) is the minimal polynomial of a. Express m(x) as a product of terms of the form (x - a). (b) Find the generating polynomial g(x) of C, expressed as a product of minimal polyno- mials m(x). (You do not need to expand each mi (r) as a product of terms of the form (x-a), other than the expansion of m(x) that you have already done in part (a).) (c) Find kdim C. 14. (24 points) Fix N = 36 and w = 2/36. Let Ho= C, H = C2, H2 = C6, H3 = C12, and H4 =C36. Recall that the main loop of the FFT based on C < C2 < C6 < C12 C36: f(0) applied to the initial input x = can be described as follows. For i = 1 to 4: [f(N-1)] == Set H-1 = (w), H = (wk), and d = m/k. d-1 Subgroup fill: For j = 0 to (N/k) 1, set y(jk) = x(jm + kr)wrkj T=0 Translate the subgroup fill to cosets of Hi, set x = y, and loop. (a) Working in terms of w, write out the elements of H2 and H3 and write out the elements of the standard transversal T2,3 for H2 in H3. (b) Write out the results of the "subgroup fill" part of step 3 (i = 3). That is, for all t corresponding to the elements of H3, write out the formula for y(t) in terms of the inputs x (the output of step 2, i = 2). (c) Draw the corresponding subgroup subdiagram for step 3 (i = 3).