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New Constructions of Reversible DNA Codes

Xueyan Chen, Whan-Hyuk Choi, Hongwei Liu

TL;DR

This work develops a general framework for constructing reversible DNA codes using composite group codes built from group rings and circulant matrices. By designing reversible composite matrices and proving reversibility results, it yields DNA codes that satisfy HD, RV, RC, and GC constraints with improved size bounds, including new lower bounds for lengths such as $80$, $96$, and $160$ at a fixed distance $d$. The approach combines algebraic tools (group rings, circulant structures) with explicit matrix constructions and computational validation in Magma, demonstrating practical gains over prior bounds. The methods offer a versatile, scalable path to large reversible DNA codes applicable to DNA data storage and computation.

Abstract

DNA codes have many applications, such as in data storage, DNA computing, etc. Good DNA codes have large sizes and satisfy some certain constraints. In this paper, we present a new construction method for reversible DNA codes. We show that the DNA codes obtained using our construction method can satisfy some desired constraints and the lower bounds of the sizes of some DNA codes are better than the known results. We also give new lower bounds on the sizes of some DNA codes of lengths $80$, $96$ and $160$ for some fixed Hamming distance $d$.

New Constructions of Reversible DNA Codes

TL;DR

This work develops a general framework for constructing reversible DNA codes using composite group codes built from group rings and circulant matrices. By designing reversible composite matrices and proving reversibility results, it yields DNA codes that satisfy HD, RV, RC, and GC constraints with improved size bounds, including new lower bounds for lengths such as , , and at a fixed distance . The approach combines algebraic tools (group rings, circulant structures) with explicit matrix constructions and computational validation in Magma, demonstrating practical gains over prior bounds. The methods offer a versatile, scalable path to large reversible DNA codes applicable to DNA data storage and computation.

Abstract

DNA codes have many applications, such as in data storage, DNA computing, etc. Good DNA codes have large sizes and satisfy some certain constraints. In this paper, we present a new construction method for reversible DNA codes. We show that the DNA codes obtained using our construction method can satisfy some desired constraints and the lower bounds of the sizes of some DNA codes are better than the known results. We also give new lower bounds on the sizes of some DNA codes of lengths , and for some fixed Hamming distance .
Paper Structure (13 sections, 8 theorems, 67 equations)

This paper contains 13 sections, 8 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. DNA Codes
  4. Circulant Matrices, Group Rings and Group Codes
  5. Composite Matrices and Composite Group Codes
  6. Reversible Composite Group Codes
  7. Reversible Group Codes
  8. Reversible Composite Group Codes
  9. Reversible Composite Matrices over Ring $R$
  10. The Case of $\frac{n}{r}=2$
  11. The Case of $\frac{n}{r}=4$
  12. Computational Results
  13. Conclusion

Key Result

Theorem 3.1

( b10) Let $R$ be a finite ring. Let $G$ be a finite group of order $n=2l$ and let $T =\{e, t_{1},t_{2},\dots,t_{l-1}\}$ be a subgroup of index $2$ in $G$. Let $\beta \in G\backslash T$ be of order $2$. List the elements of $G$ as in eq.3.1, then any linear $G$-code in $R^{n}$ (a left ideal in $RG$

Theorems & Definitions (23)

  • Definition 2.1
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 3.1
  • Example 3.1
  • Theorem 3.1
  • ...and 13 more