DNA codes from $( ext{ extbaro}, mathfrak{d}, gamma)$-constacyclic codes over $mathbb{Z}_4+omegamathbb{Z}_4$

2024-12-13

AI Summary

This paper addresses the construction of DNA codes over the non-chain ring $mathbb{Z}_4 + omegamathbb{Z}_4$ satisfying reverse-complement constraints, motivated by emerging requirements in DNA computing and storage. Method: We introduce, for the first time, a $(ar{o}, mathfrak{d}, gamma)$-constacyclic code framework equipped with a ring automorphism and a derivation, providing a complete characterization of generator polynomials of arbitrary length and establishing necessary and sufficient conditions for reverse-complement closure. Two reversible and optimal DNA code constructions are proposed, integrating the Gray map and a bijective mapping between DNA bases (A/T/C/G) and ring elements (0/1/2/3). Contribution/Results: Our approach yields multiple new optimal DNA codes and improved $mathbb{Z}_4$-linear codes whose parameters surpass those of previously known codes in the z4codes database, significantly advancing the constructive theory and applicability of ring-based DNA coding.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Distributed Search

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

This work introduces a novel approach to constructing DNA codes from linear codes over a non-chain extension of $mathbb{Z}_4$. We study $( ext{ extbaro},mathfrak{d}, gamma)$-constacyclic codes over the ring $mathfrak{R}=mathbb{Z}_4+omegamathbb{Z}_4, omega^2=omega,$ with an $mathfrak{R}$-automorphism $ ext{ extbaro}$ and a $ ext{ extbaro}$-derivation $mathfrak{d}$ over $mathfrak{R}.$ Further, we determine the generators of the $( ext{ extbaro},mathfrak{d}, gamma)$-constacyclic codes over the ring $mathfrak{R}$ of any arbitrary length and establish the reverse constraint for these codes. Besides the necessary and sufficient criterion to derive reverse-complement codes, we present a construction to obtain DNA codes from these reversible codes. Moreover, we use another construction on the $( ext{ extbaro},mathfrak{d},gamma)$-constacyclic codes to generate additional optimal and new classical codes. Finally, we provide several examples of $( ext{ extbaro},mathfrak{d}, gamma)$ constacyclic codes and construct DNA codes from established results. The parameters of these linear codes over $mathbb{Z}_4$ are better and optimal according to the codes available at cite{z4codes}.

Problem

• Construct DNA codes from linear codes
• Determine generators of constacyclic codes
• Establish reverse constraint for DNA codes

Innovation

• Construct DNA codes from linear codes
• Use $( ext{ extbaro},mathfrak{d}, γ)$-constacyclic codes
• Establish reverse-complement codes constraints