# Skew Cyclic Codes Over $\mathbb{F}_4 R$

## TL;DR

This paper introduces skew cyclic codes over a new ring $\\mathbb{F}_4 R$, explores their algebraic properties, and connects them to DNA code design with biomolecular constraints.

## Contribution

It characterizes $\\mathbb{F}_4 R$-skew cyclic codes, defines a nondegenerate inner product for self-orthogonality, and links these codes to DNA coding applications.

## Key findings

- Characterization of $\\mathbb{F}_4 R$-skew cyclic codes.
- Connections between Gray map images and linear cyclic codes over $\\mathbb{F}_4$.
- Identification of reversible complement skew cyclic codes.

## Abstract

This paper considers a new alphabet set, which is a ring that we call $\mathbb{F}_4R$, to construct linear error-control codes. Skew cyclic codes over the ring are then investigated in details. We define a nondegenerate inner product and provide a criteria to test for self-orthogonality. Results on the algebraic structures lead us to characterize $\mathbb{F}_4R$-skew cyclic codes. Interesting connections between the image of such codes under the Gray map to linear cyclic and skew-cyclic codes over $\mathbb{F}_4$ are shown. These allow us to learn about the relative dimension and distance profile of the resulting codes. Our setup provides a natural connection to DNA codes where additional biomolecular constraints must be incorporated into the design. We present a characterization of $R$-skew cyclic codes which are reversible complement.