New DNA Codes from Cyclic Codes over Mixed Alphabets

Hai Q. Dinh, Sachin Pathak, Ashish Kumar Upadhyay and Woraphon Yamaka
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Hai Q. Dinh: Department of Mathematics, Kent State University, Warren, OH 44483, USA
Sachin Pathak: Department of Mathematics, Indian Institute of Technology Patna, Patna 801103, India
Ashish Kumar Upadhyay: Department of Mathematics, Indian Institute of Technology Patna, Patna 801103, India
Woraphon Yamaka: Centre of Excellence in Econometrics, Faculty of Economics, Chiang Mai University, Chiang Mai 50200, Thailand

Mathematics, 2020, vol. 8, issue 11, 1-24

Abstract: Let R = F 4 + u F 4 , with u 2 = u and S = F 4 + u F 4 + v F 4 , w i t h u 2 = u , v 2 = v , u v = v u = 0 . In this paper, we study F 4 R S -cyclic codes of block length ( α , β , γ ) and construct cyclic DNA codes from them. F 4 R S -cyclic codes can be viewed as S [ x ] -submodules of F q [ x ] 〈 x α − 1 〉 × R [ x ] 〈 x β − 1 〉 × S [ x ] 〈 x γ − 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 ψ 1 from R to F 4 2 and ψ 2 from S to F 4 3 , 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 F 4 , R , S and F 4 R S to be reversible and reverse-complement. As applications, we provide examples of new cyclic DNA codes constructed by our results.

Keywords: cyclic codes; reversible codes; reversible-complement codes; cyclic DNA codes (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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