The trace dual of nonlinear skew cyclic codes
This work addresses the problem of characterizing dual codes for a class of codes used in quantum error correction, but the results are incremental as they extend known dual code formulas to skew cyclic codes.
The paper derives generators for trace Euclidean and trace Hermitian dual codes of general $\mathbb{F}_q$-linear cyclic and skew cyclic codes over $\mathbb{F}_{q^2}$, for odd prime power $q$. This provides a theoretical foundation for constructing quantum error-correcting codes from these families.
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 $\mathbb{F}_q\leq \mathbb{F}_{q^2}$ 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 $\mathbb{F}_{q^2}[X;σ]/(X^n-1)$, where $σ$ denotes the Frobenius automorphism and $\mathbb{F}_{q^2}[X;σ]$ the induced skew polynomial ring.