<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ q $\end{document}</tex-math></inline-formula> be an odd prime power and <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{F}_q $\end{document}</tex-math></inline-formula> be the finite field with <inline-formula><tex-math id="M3">\begin{document}$ q $\end{document}</tex-math></inline-formula> elements. In this paper, suppose ring <inline-formula><tex-math id="M4">\begin{document}$ R = \mathbb{F}_{q}+ \mu \mathbb{F}_{q}+ \nu \mathbb{F}_{q}+ \mu \nu \mathbb{F}_{q} $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M5">\begin{document}$ \mu \nu = \nu \mu, \mu^{2} = \mu, \nu^{2} = \nu. $\end{document}</tex-math></inline-formula> We first give a Gray map from <inline-formula><tex-math id="M6">\begin{document}$ R $\end{document}</tex-math></inline-formula> onto <inline-formula><tex-math id="M7">\begin{document}$ \mathbb{F}_q^{4} $\end{document}</tex-math></inline-formula> and consider a decomposition of the ring <inline-formula><tex-math id="M8">\begin{document}$ R $\end{document}</tex-math></inline-formula>. Additionally, we investigate linear complementary dual (LCD) codes over the ring <inline-formula><tex-math id="M9">\begin{document}$ R $\end{document}</tex-math></inline-formula>. Some conditions for such linear codes over <inline-formula><tex-math id="M10">\begin{document}$ R $\end{document}</tex-math></inline-formula> to be linear complementary dual are given. Furthermore, based on the Artin conjecture, we get a class of good codes by calculating the total number of LCD double circulant codes over <inline-formula><tex-math id="M11">\begin{document}$ R $\end{document}</tex-math></inline-formula>.</p>
Let [Formula: see text] be an odd prime number, [Formula: see text] for a positive integer [Formula: see text], let [Formula: see text] be the finite field with [Formula: see text] elements and [Formula: see text] be a primitive element of [Formula: see text]. We first give an orthogonal decomposition of the ring [Formula: see text], where [Formula: see text] and [Formula: see text] for a fixed integer [Formula: see text]. In addition, Galois dual of a linear code over [Formula: see text] is discussed. Meanwhile, constacyclic codes and cyclic codes over the ring [Formula: see text] are investigated as well. Remarkably, we obtain that if linear codes [Formula: see text] and [Formula: see text] are a complementary pair, then the code [Formula: see text] and the dual code [Formula: see text] of [Formula: see text] are equivalent to each other.
In this paper, we investigate the algebraic structure of [Formula: see text]-additive codes, where [Formula: see text] and [Formula: see text] are nonnegative integers, [Formula: see text] (respectively, [Formula: see text]) denotes the finite field of order 2 (respectively, [Formula: see text]). We first give the generator polynomials of additive cyclic codes over [Formula: see text] and then the generator polynomials of additive cyclic codes over [Formula: see text] is also given. In addition, we introduce a linear map [Formula: see text], and study its properties. What’s more, the dual of additive cyclic codes over [Formula: see text] are investigated as well. And we get that the duals of any additive cyclic codes over [Formula: see text] are also additive cyclic codes. Finally, separable [Formula: see text]-additive cyclic codes are investigated.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with đź’™ for researchers
Part of the Research Solutions Family.