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

Year 2023, Volume: 33 Issue: 33, 77 - 86, 09.01.2023

R. Nekooeı Z. Pourshafıey

https://doi.org/10.24330/ieja.1198011

Abstract

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.

Keywords

Cyclic code, quasi-cyclic code, additive cyclic code, linear code

References

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