5. (a) Suppose that a binary linear code C generated by g(x) has the property that whenever c(x) is a codeword, then so is x"-c(x ]) - which is the same codeword with the bits written in reverse order. Such a code is called reversible. Prove that g(x) = xg(x ]) in this case, where r = degg(x). (b) What is the corresponding statement for reversible nonbinary cyclic codes? (c) Show that a cyclic code is reversible if and only if g(7) = 0 implies g(7 ]) = 0 for any n-th root of unity y. (d) Show that if there exists an integer m such that q = -1 mod n, then every cyclic code of length n over GF(q) is reversible. 6. A Vandermonde matrix is a square m x m matrix of the form . . H am am Vm = am . . . . . . . . . . . . am-1 am-1 . . m-1 a'm where a1, a2, . .. 'm are elements in a field F (not necessarily finite). Prove that m-1 m det( Vm) = II II ( xi - x ;) j=1i=j+1 and hence Vm is nonsingular if and only if a1, a2, . .. am are all distinct