7 research outputs found

    Skew cyclic codes over Z4+vZ4\mathbb{Z}_4+v\mathbb{Z}_4 with derivation: structural properties and computational results

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    In this work, we study a class of skew cyclic codes over the ring R:=Z4+vZ4,R:=\mathbb{Z}_4+v\mathbb{Z}_4, where v2=v,v^2=v, with an automorphism θ\theta and a derivation Δθ,\Delta_\theta, namely codes as modules over a skew polynomial ring R[x;θ,Δθ],R[x;\theta,\Delta_{\theta}], whose multiplication is defined using an automorphism θ\theta and a derivation Δθ.\Delta_{\theta}. We investigate the structures of a skew polynomial ring R[x;θ,Δθ].R[x;\theta,\Delta_{\theta}]. We define Δθ\Delta_{\theta}-cyclic codes as a generalization of the notion of cyclic codes. The properties of Δθ\Delta_{\theta}-cyclic codes as well as dual Δθ\Delta_{\theta}-cyclic codes are derived. As an application, some new linear codes over Z4\mathbb{Z}_4 with good parameters are obtained by Plotkin sum construction, also via a Gray map as well as residue and torsion codes of these codes.Comment: 25 page

    On DNA Codes Over the Non-Chain Ring Z4+uZ4+u2Z4\mathbb{Z}_4+u\mathbb{Z}_4+u^2\mathbb{Z}_4 with u3=1u^3=1

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    In this paper, we present a novel design strategy of DNA codes with length 3n3n over the non-chain ring R=Z4+uZ4+u2Z4R=\mathbb{Z}_4+u\mathbb{Z}_4+u^2\mathbb{Z}_4 with 6464 elements and u3=1u^3=1, where nn denotes the length of a code over RR. We first study and analyze a distance conserving map defined over the ring RR into the length-33 DNA sequences. Then, we derive some conditions on the generator matrix of a linear code over RR, which leads to a DNA code with reversible, reversible-complement, homopolymer 22-run-length, and w3n\frac{w}{3n}-GC-content constraints for integer ww (0≤w≤3n0\leq w\leq 3n). Finally, we propose a new construction of DNA codes using Reed-Muller type generator matrices. This allows us to obtain DNA codes with reversible, reversible-complement, homopolymer 22-run-length, and 23\frac{2}{3}-GC-content constraints.Comment: This paper has been presented in IEEE Information Theory Workshop (ITW) 2022, Mumbai, INDI

    Reversible DNA codes from skew cyclic codes over a ring of order 256

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    We introduce skew cyclic codes over the finite ring R\R, where u2=0,v2=v,w2=w,uv=vu,uw=wu,vw=wvu^{2}=0,v^{2}=v,w^{2}=w,uv=vu,uw=wu,vw=wv and use them to construct reversible DNA codes. The 4-mers are matched with the elements of this ring. The reversibility problem for DNA 4-bases is solved and some examples are provided

    Kernel Code for DNA Digital Data Storage

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    The biggest challenge when using DNA as a storage medium is maintaining its stability. The relative occurrence of Guanine (G) and Cytosine (C) is essential for the longevity of DNA. In addition to that, reverse complementary base pairs should not be present in the code. These challenges are overcome by a proper choice of group homomorphisms. Algorithms for storage and retrieval of information in DNA stings are written by using kernel code. Complexities of these algorithms are less compared to the existing algorithms. Construction procedures followed in this paper are capable of constructing codes of required sizes and Reverse complement distance.Comment: 12 pages, 1 figur
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