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