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New constructions of DNA codes under multiple constraints and parallel searching algorithms

Guodong Wang, Hongwei Liu, Xueyan Chen

TL;DR

This paper extends the results of constructing reversible group codes and reversible composite group codes to general even-order finite groups, and establishes a relationship between the GC-weight enumerator of DNA codes and the Hamming weight enumerator of their trace codes, which greatly improves the computational efficiency of searching for DNA codes.

Abstract

DNA codes have garnered significant interest due to their utilization in digital media storage, cryptography, and DNA computing. In this paper, we first extend the results of constructing reversible group codes \cite{Cengellenmis} and reversible composite group codes \cite{Korban5} to general even-order finite groups. By using these results, we give parallel searching algorithms to find some new DNA codes with better parameters. Secondly, by mapping codes over $\mathbb{F}_4$ to DNA codes, we establish a relationship between the $GC$-weight enumerator of DNA codes and the Hamming weight enumerator of their trace codes, which greatly improves the computational efficiency of searching for DNA codes. Based on this relationship, we propose an efficient algorithm for generating DNA codes with $50\%$ $GC$-content. Furthermore, we find that there is no direct connection between the $GC$-weight enumerator of a DNA code and the $GC$-weight enumerator of its dual code. Finally, we present algorithms for determining whether a DNA code is free from secondary structures or conflict-free, and some new DNA codes with better parameters under multiple constraints are obtained, which are listed in Tables 1 and 4.

New constructions of DNA codes under multiple constraints and parallel searching algorithms

TL;DR

This paper extends the results of constructing reversible group codes and reversible composite group codes to general even-order finite groups, and establishes a relationship between the GC-weight enumerator of DNA codes and the Hamming weight enumerator of their trace codes, which greatly improves the computational efficiency of searching for DNA codes.

Abstract

DNA codes have garnered significant interest due to their utilization in digital media storage, cryptography, and DNA computing. In this paper, we first extend the results of constructing reversible group codes \cite{Cengellenmis} and reversible composite group codes \cite{Korban5} to general even-order finite groups. By using these results, we give parallel searching algorithms to find some new DNA codes with better parameters. Secondly, by mapping codes over to DNA codes, we establish a relationship between the -weight enumerator of DNA codes and the Hamming weight enumerator of their trace codes, which greatly improves the computational efficiency of searching for DNA codes. Based on this relationship, we propose an efficient algorithm for generating DNA codes with -content. Furthermore, we find that there is no direct connection between the -weight enumerator of a DNA code and the -weight enumerator of its dual code. Finally, we present algorithms for determining whether a DNA code is free from secondary structures or conflict-free, and some new DNA codes with better parameters under multiple constraints are obtained, which are listed in Tables 1 and 4.
Paper Structure (9 sections, 12 theorems, 46 equations, 4 tables, 4 algorithms)

This paper contains 9 sections, 12 theorems, 46 equations, 4 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Group codes and composite matrix group codes
  4. $GC$-weight enumerator of DNA codes
  5. Algorithms
  6. Finding of DNA codes
  7. $50\%$ $GC$-content constraint
  8. Other specific constraints
  9. Conclusion and further research

Key Result

Theorem 3.2

Cengellenmis Let $G$ be a finite group of order $n=2l$ and $H= \{ e, h_{1}, h_{2}, \ldots , h_{l- 1}\}$ be a subgroup of index $2$ in $G$. Let $\beta \notin H$ be an element in $G$ with $\beta ^{- 1}=\beta$. List the elements of $G$ as then any linear $G$-code in $R^n$ (a left ideal in $RG$) is a reversible code.

Theorems & Definitions (23)

  • Definition 3.1
  • Theorem 3.2
  • Definition 3.3
  • Theorem 3.4
  • Remark 3.5
  • Theorem 3.6
  • proof
  • Corollary 3.7
  • proof
  • Example 3.8
  • ...and 13 more