Skew-Cyclic and Constacyclic Codes over Finite Chain Rings for DNA-Based Data Storage and Error Correction

Abstract

DNA has considerable potential for long-term data storage, but the process of synthesising, preserving, and sequencing DNA strands can introduce substitutions, insertions, and deletions. In addition, biochemical requirements restrict the sequences that can be used reliably. This paper develops a structured framework for constructing error-correcting DNA codes from skew-cyclic and skew-constacyclic codes over finite chain rings. We first select a chain ring whose additive structure can represent the four DNA nucleotides and define a nucleotide map under which Watson–Crick complementation corresponds to addition by a fixed ring element. We then introduce the skew polynomial ring  and describe skew -constacyclic codes as left ideals of . Their generator representations and cardinalities are presented alongside sufficient conditions for reversibility, reverse-complement closure, and balanced GC-content. Error performance is considered through Lee weight and a distance-preserving Gray map. Finally, we outline an end-to-end DNA storage workflow and discuss illustrative constructions: a length-four code over  and a genuinely skew construction over  that demonstrates non-unique factorisation. The examples clarify the algebraic framework; practical biochemical performance remains a matter for further testing.