Journal of Applied Mathematics and Computing

$$\pmb {{\mathbb {F}}}_q$$-Linear skew cyclic codes over $$\pmb {{\mathbb {F}}}_{q^2}$$ and their applications of quantum codes construction

Yun Gao, Jian Gao, Shilin Yang, Fang‐Wei Fu

Published Mar 29, 2021• 0 citations

Publisher

DOI: 10.1007/s12190-020-01479-5

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\pmb {{\mathbb {F}}}_q

-Linear skew cyclic codes over

\pmb {{\mathbb {F}}}_{q^2}

and their applications of quantum codes construction

{\mathbb {F}}_qR

-linear skew constacyclic codes and their application of constructing quantum codes

Entanglement-assisted binary quantum codes from skew cyclic codes over

{\mathbb {F}}_2 \times ({\mathbb {F}}_2+v{\mathbb {F}}_2)

\mathbb{F}_q\mathcal{R}

-skew cyclic codes and their application to quantum codes

Let

p

be a prime and

\mathbb{F}_q

be the finite field of order

q=p^m

. In this paper, we study

\mathbb{F}_q\mathcal{R}

-skew cyclic codes where

\mathcal{R}=\mathbb{F}_q+u\mathbb{F}_q

with

u^2=u

. To characterize

\mathbb{F}_q\mathcal{R}

-skew cyclic codes, we first establish their algebraic structure and then discuss the dual-containing properties by considering a non-degenerate inner product. Further, we define a Gray map over

\mathbb{F}_q\mathcal{R}

and obtain their

\mathbb{F}_q

-Gray images. As an application, we apply the CSS (Calderbank-Shor-Steane) construction on Gray images of dual containing

\mathbb{F}_q\mathcal{R}

-skew cyclic codes and obtain many quantum codes with better parameters than the best-known codes available in the literature.