A relation between quasi-cyclic codes and 2-D cyclic codes

We consider a q-ary quasi-cyclic code C of length mℓ and index ℓ, where both m and ℓ are relatively prime to q. If the constituents of C are cyclic codes, we show that C can also be viewed as a 2-D cyclic code of size m×ℓ over Fq. In case m and ℓ are also coprime to each other, we easily observe that the code C must be equivalent to a cyclic code, which was proved earlier by Lim.