The trace dual of nonlinear skew cyclic codes

Abstract

Codes which have a finite field ${\mathbb{F}}_{q^m}$ as their alphabet but which are only linear over a subfield ${\mathbb{F}}_q$ are a topic of much recent interest due to their utility in constructing quantum error correcting codes. In this article, we find generators for trace dual spaces of different families of ${\mathbb{F}}_q$-linear codes over ${\mathbb{F}}_{q^2}$. In particular, given the field extension ${\mathbb{F}}_q\le {\mathbb{F}}_{q^2}$ with $q$ an odd prime power, we determine the trace Euclidean and trace Hermitian dual codes for the general ${\mathbb{F}}_q$-linear cyclic ${\mathbb{F}}_{q^2}$-code. In addition, we also determine the trace Euclidean and trace Hermitian duals for general ${\mathbb{F}}_q$-linear skew cyclic ${\mathbb{F}}_{q^2}$-codes, which are defined to be left ${\mathbb{F}}_q[X]$-submodules of ${\mathbb{F}}_{q^2}[X;\ddot{I}ƒ]/(X^n-1)$, where $\ddot{I}ƒ$ denotes the Frobenius automorphism and ${\mathbb{F}}_{q^2}[X;\ddot{I}ƒ]$ the induced skew polynomial ring.

17 pages