Abstract
Let R=F4+uF4,withu2=u and S=F4+uF4+vF4,withu2=u,v2=v,uv=vu=0. In this paper, we study F4RS-cyclic codes of block length (α,β,γ) and construct cyclic DNA codes from them. F4RS-cyclic codes can be viewed as S[x]-submodules of Fq[x]⟨xα−1⟩×R[x]⟨xβ−1⟩×S[x]⟨xγ−1⟩. We discuss their generator polynomials as well as the structure of separable codes. Using the structure of separable codes, we study cyclic DNA codes. By using Gray maps ψ1 from R to F42 and ψ2 from S to F43, we give a one-to-one correspondence between DNA codons of the alphabets {A,T,G,C}2,{A,T,G,C}3 and the elements of R,S, respectively. Then we discuss necessary and sufficient conditions of cyclic codes over F4, R, S and F4RS to be reversible and reverse-complement. As applications, we provide examples of new cyclic DNA codes constructed by our results.
Highlights
Linear codes were introduced in the late 1940s by Shannon and have a central role in InformationTheory for recovering the corrupted messages that are sent through a noisy communication channel.Initially, linear codes were studied over finite fields, but in the early 1970s, these codes were discussed over finite rings
We provide examples of new cyclic Deoxyribonucleic acid (DNA) codes constructed by our results
Motivated by the idea of the construction of LCD codes and quantum codes over mixed alphabets, in this paper, we study the construction of cyclic DNA codes over single alphabets
Summary
Linear codes were introduced in the late 1940s by Shannon and have a central role in Information. By following the approach of Abualrub et al [7], many researchers have discussed the structure of cyclic and constacyclic codes over mixed alphabets and determined their generators as well as minimal generating sets [8,9,10,11]. Liang et al [20] studied cyclic DNA codes over the same ring and discussed some necessary and sufficient conditions for reversible and reversible-complement codes.
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