Skew cyclic codes over 𝔽4R

Abstract

This paper considers a new alphabet set, which is a ring that we call [Formula: see text], to construct linear error-control codes. Skew cyclic codes over this 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 [Formula: see text]-skew cyclic codes. Interesting connections between the image of such codes under the Gray map to linear cyclic and skew-cyclic codes over [Formula: see text] 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 [Formula: see text]-skew cyclic codes which are reversible complement.