Abstract
In this paper we study the structure and properties of additive right and left polycyclic codes induced by a nonbinary vector \(\textbf{a}\in \mathbb {F} _{4}^{n}\), where \(\mathbb {F}_{4}\) is the finite field of order 4. We show that additive right and left polycyclic codes are \(\mathbb {F}_{2}[x]\) -submodules of the rings \(R_{n}=\mathbb {F}_{4}\left[ x\right] /\left\langle x^{n}-a\left( x\right) \right\rangle \) and \(S_{n}=\mathbb {F}_{4}\left[ x \right] /\left\langle x^{n}-a_{r}\left( x\right) \right\rangle \) respectively. We also show that these codes are invariant under multiplication by a certain matrix D, and construct their generator polynomials. Moreover, we study the relationship between additive polycyclic codes and linear polycyclic codes. We identify cases in which additive right polycyclic codes are linear right polycyclic codes and other cases in which additive right polycyclic codes are not linear right polycyclic codes. Finally, we give some applications of these codes by constructing examples of codes with good parameters.
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Abualrub, T., Soufi Karbaski, A., Aydin, N. et al. Additive polycyclic codes over \(\mathbb {\pmb {\varvec{F}}}_{4}\) induced by nonbinary polynomials. J. Appl. Math. Comput. 69, 4821–4832 (2023). https://doi.org/10.1007/s12190-023-01940-1
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DOI: https://doi.org/10.1007/s12190-023-01940-1