The q-ary image of a qm-ary cyclic code

• Authors: G. E. Seguin
• Venue: IEEE Transactions on Information Theory
• Year: 2006

For (n, q)=1 V a qm-ary cyclic code of length n and with generator polynomial g(x), we show that there exists a basis for F(qm) over Fq with respect to which the q-ary image of V is cyclic, if and only if: (i) g(x) is over Fq; or (ii) g(x)=g0(x)(x-γ-q(μ)), g0(x) is over Fq, Fq≠F(qk)=Fq(γ)⊂F(qm ), μ an integer modulo k, and wm-γ has a divisor over F(qk) of degree e=m/k; or (iii) g(x)=g0 (x) Πμεs(x-γ(-qμ)), g 0(x) is over Fq, Fq≠F(qk)=Fq(γ)⊂F(qm ), S a set of integers module k of cardinality k-1 and wm -μ 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