Reversible and Reversible-Complement Double Cyclic Codes over F4+vF4 and its Application to DNA Codes

Abstract: In this article, we study the algebraic structure of double cyclic codes of length $(m, n)$ over $\mathbb{F}4$ and we give a necessary and sufficient condition for a double cyclic code over $\mathbb{F}_4$ to be reversible. Also, we determine the algebraic structure of double cyclic codes of length $(m, n)$ over $\mathbb{F}_4+v\mathbb{F}_4$ with $v^2=v$, satisfying the reverse constraint and the reverse-complement constraint. Then we establish a one-to-one correspondence $ψ$ between the 16 DNA double pairs $S{D_{16}} $ and the 16 elements of the finite ring $\mathbb{F}_4+v\mathbb{F}_4$. We also discuss the GC-content of DNA double cyclic codes.

• The paper establishes an algebraic framework for double cyclic codes over F4+vF4 and provides criteria for reversibility and reversible-complement properties.
• It leverages module decompositions and generator polynomials to translate cyclic code structures into Watson-Crick-compliant DNA double pairs.
• The study offers practical methods for analyzing GC-content and optimizing DNA code designs for applications in computational biology and molecular storage.

Reversible and Reversible-Complement Double Cyclic Codes over $\mathbb{F}_4 + v\mathbb{F}_4$ and DNA Applications

Introduction

This work presents a comprehensive algebraic framework for reversible and reversible-complement double cyclic codes over the finite non-chain ring $\mathbb{F}_4 + v\mathbb{F}_4$ with $v^2 = v$. The motivation arises from a synergy between algebraic coding theory and the design of DNA codes, where constraints related to reversibility and Watson-Crick complementarity are crucial for DNA computing and molecular storage systems.

Algebraic Structure of Double Cyclic Codes

Double cyclic codes extend conventional cyclic codes by considering a coordinate partition and enforcing cyclicity independently on each part. The paper provides a structural characterization for double cyclic codes of length $(m,n)$ over both $\mathbb{F}_4$ and $\mathcal{R} = \mathbb{F}_4 + v\mathbb{F}_4$, exploiting the module and ideal structures induced by the coordinate-wise cyclic shifts.

Over $\mathbb{F}_4$, these codes are $\mathbb{F}_4[x]$-submodules of the product ring $$\mathcal{A}_{m,n} = \mathbb{F}_4[x]/\langle x^m-1\rangle \times \mathbb{F}_4[x]/\langle x^n-1\rangle$$ under the component-wise action. The generating set for such double cyclic codes is expressed in terms of generator polynomials $b(x), l(x), a(x)$ satisfying certain divisibility and degree conditions. Extending this to $\mathcal{R}$, results from [bathala2021some] are employed to demonstrate that every double cyclic code over $\mathcal{R}$ decomposes as $(1+v)\mathcal{C}_1 \oplus v\mathcal{C}_2$, with $\mathcal{C}_1$ and $\mathcal{C}_2$ being double cyclic codes over $\mathbb{F}_4$.

Reversibility and Reversible-Complement Constraints

The reversibility of a code is central in DNA coding theory to facilitate certain biological operations and error resilience. The paper establishes necessary and sufficient algebraic criteria for an $\mathbb{F}_4$-double cyclic code of length $(m,n)$ to be reversible, involving self-reciprocity of the generator polynomials and further constraints linking the generators. These algebraic conditions are then lifted via module decompositions to provide criteria for reversibility over $\mathcal{R}$.

Additionally, the reverse-complement property, critical for modeling Watson-Crick complementarity, is investigated. It is proven that a double cyclic code over $\mathcal{R}$ is reversible-complement if and only if it is reversible and contains the all-one codeword. Technical lemmas regarding additive and complement operations in $\mathcal{R}$ underpin the proof, reflecting the interaction between code structure and biological admissibility.

DNA Code Construction and Map to Double Pairs

A key contribution is the explicit construction of a bijection $\psi$ between the 16 elements of $\mathbb{F}_4 + v\mathbb{F}_4$ and the set $S_{D_{16}}$ of DNA double pairs. This mapping is formulated in such a way that biologically relevant Watson-Crick complementarity is mirrored by algebraic complement operation in the ring. Through a secondary mapping $\theta$, double cyclic codes over $\mathcal{R}$ give rise to codes in DNA double pair alphabets, enabling the translation of algebraic code-theoretic constraints into properties of DNA words.

Correspondence tables for element-complement and the DNA double pairs secure the preservation of biological constraints under the ring-to-DNA mapping, making these constructions directly applicable to biological coding applications and facilitating the encoding of sequences with specified error tolerance, chemical stability, and hybridization properties.

GC Content Analysis

GC-content uniformity is essential for DNA word design to ensure controlled melting temperature and error rates. The article characterizes the GC-content of DNA codes derived from $\mathcal{R}$-double cyclic codes by means of an explicit enumerator formula, relating it to Hamming weights of codewords and their images under the extended Gray map. This enables the direct computation and optimization of code parameters for practical molecular implementations.

Implications and Future Directions

The presented results directly inform the systematic construction of large classes of DNA codes satisfying stringent algebraic and biophysical constraints, including reversibility and fixed GC-content, which are critical for DNA-based computation and storage systems. The algebraic structure over $\mathbb{F}_4 + v\mathbb{F}_4$ leverages the rich module theory of non-chain Frobenius rings, suggesting further investigation into optimal code parameters (distance, size, weight distribution) and their biological trade-offs.

Potential avenues for extension include:

• Examination of the automorphism group structure of these codes for further code class classification;
• Algorithm development for efficient encoding and decoding over $\mathbb{F}_4 + v\mathbb{F}_4$;
• Generalization to more complex ring structures and higher-arity DNA alphabets;
• Application to constraint satisfaction for combinatorial DNA libraries with additional biochemical constraints.

Conclusion

This paper rigorously characterizes reversible and reversible-complement double cyclic codes over $\mathbb{F}_4 + v\mathbb{F}_4$, elucidating their algebraic underpinnings and establishing concrete connections to Watson-Crick-compliant DNA coding schemes. The algebraic correspondences and enumerative results provide a principled framework for designing DNA codes with properties matching practical requirements in both computational biology and information theory contexts (2512.13295).