Q4:

(a) Construct the finite field of order 16 using a 5-term irreducible quartic poly- nomial. List the elements of the field in a four-column table, with the fourth column having the vector form of the elements, the third column having the polynomial form in terms of a primitive element , the second column having the exponential form in terms of where possible and the first column having the discrete logarithms of the terms in the second column (when they exist). Explain how you have produced the polynomial form from each exponential form where the exponent is greater than 4.

(b) Construct the cyclotomic cosets of 2 modulo 15.

(c) For each cyclotomic coset, multiply together the corresponding linear poly-

nomials over F16 so as to obtain an irreducible polynomial over Z2.

(d) How many different binary polynomials of degree 5 that divide x15 + 1 can be used to construct cyclic codes with minimum distance at least 4? [Justify your answer.]