WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 23, 2024
On Quantum Codes over Non Local Rings
Authors: , ,
Abstract: We study the structural properties of the ring $$\Re= \mathbb{F}_{p}+u\mathbb{F}_{p}+u^{2}\mathbb{F}_{p}+u^{3}\mathbb{F}_{p}$$, where $$p\neq2$$ is a prime
and $$u^{4}=u^{3}$$ , as well as the linear codes over R. We investigate the generator’s cyclic codes and their dual codes
over $$R$$. An isometric Gray map from $$R$$ to $$\mathbb{F}_{p}^{4}$$ is defined. We offer an equivalence condition that cyclic codes must
satisfy over $$R$$ in order to include their dual. Moreover, we establish the existence of quantum error-correcting
codes based on cyclic codes over $$R$$. Finally, under various criteria such as cyclic codes length and generator
polynomial degree, we build quantum error-correcting codes over $$\Re$$ which their dimensions are divided by $$p$$ and $$2^{4}$$.