THE Q-ARY IMAGE OF A Q(M)-ARY CYCLIC CODE

For (n, q) = 1 V a q(m)-ary cyclic code of length n and with generator polynomial g(x), we show that there exists a basis for F-qm Over F-q with respect to which the q-ary image of V is cyclic, if and only if: i) g(x) is over F-q; or ii) g(x) = g(0)(x)(x - gamma(-q mu)), g(0)(x) is over F-q, F-q not equal F-qk = F-q(gamma) subset of F-qm, mu an integer module k, and w(m) - gamma has a divisor over F-qk of degree e = m/k; or iii) g(x) = g(0)(x) Pi(mu is an element of S)(x - y(-q mu)), g(0)(x) is over F-q, F-q not equal F-qk = F-q(gamma) subset of F-qm, S a set of integers modulo k of cardinality k - 1 and w(m) - gamma has a divisor over F-qk of degree e = m/k. In all of the above cases, we determine all of the bases with respect to which the q-ary image of V is cyclic.