DNA codes from $(\text{\textbaro}, \mathfrak{d}, γ)$-constacyclic codes over $\mathbb{Z}_4+ω\mathbb{Z}_4$
Priyanka Sharma, Ashutosh Singh, Om Prakash
TL;DR
This paper develops a skew-constacyclic coding framework over the non-chain ring $\mathfrak{R}=\mathbb{Z}_4+\omega\mathbb{Z}_4$ to generate DNA codes. It defines $(\textbaro,\mathfrak{d},\gamma)$-CC codes as left modules over the skew polynomial ring $\mathfrak{R}[x;\textbaro,\mathfrak{d}]$, derives generator structures, and establishes $R$- and $RC$-constraints. DNA-code construction is achieved via the Gray map, with two constructions including code-sum methods, yielding both DNA codes and improved $\mathbb{Z}_4$-linear codes. The results provide several new and optimal codes and offer a pathway for practical DNA data storage codes and further algebraic constructions.
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 $(\text{\textbaro},\mathfrak{d}, γ)$-constacyclic codes over the ring $\mathfrak{R}=\mathbb{Z}_4+ω\mathbb{Z}_4, ω^2=ω,$ with an $\mathfrak{R}$-automorphism $\text{\textbaro}$ and a $\text{\textbaro}$-derivation $\mathfrak{d}$ over $\mathfrak{R}.$ Further, we determine the generators of the $(\text{\textbaro},\mathfrak{d}, γ)$-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 $(\text{\textbaro},\mathfrak{d},γ)$-constacyclic codes to generate additional optimal and new classical codes. Finally, we provide several examples of $(\text{\textbaro},\mathfrak{d}, γ)$ 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}.