When do quasi-cyclic codes have $\mathbb F_{q^l}$-linear image?

A length $ml$, index $l$ quasi-cyclic code can be viewed as a cyclic of $m$ over the field $\mathbb F_{q^l}$ via basis extension F_{q^l}/\mathbb F_{q}$. This is an additive code. In [C. Güneri, F. Özdemir, P. Solé, On structure codes, Discrete. Math., 341 (2018), 2735-2741], authors characterize $(l,m)$ values for one-generator codes which it impossible to have F_{q^l}$-linear image any choice polynomial But this characterization some very intricate. paper, by use characterization, we give more simple characterization.