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

Yıl 2023, Cilt: 33 Sayı: 33, 77 - 86, 09.01.2023

R. Nekooeı Z. Pourshafıey

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

Öz

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.

Anahtar Kelimeler

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

Kaynakça

• J. Bierbrauer, The theory of cyclic codes and a generalization to additive codes, Des.Codes Cryptogr., 25(2) (2002), 189-206.
• C. Güneri, F. Özdemir and P. Sole, On the additive cyclic structure of quasi-cycliccodes, Discrete. Math., 341(10) (2018), 2735-2741.
• S. Ling and C. Xing, Coding Theory, Cambridge University Press, 2004.
• M. Shi, J. Tang, M. Ge, L. Sok and P. Sole, A special class ofquasi-cyclic codes, Bull. Aust. Math. Soc., 96(3) (2017), 513-518.
• M. Shi, R. Wu and P. Sole, Long cyclic codes are good, arXiv: 1709.09865v3 [cs.IT], 17 oct 2017, 1-5.