Construction of Quantum Codes over the Class of Commutative Rings and Their Applications to DNA Codes

Shakir Ali (), Amal S. Alali, Elif Segah Oztas and Pushpendra Sharma
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Shakir Ali: Department of Mathematics, Faculty of Science, Aligarh Muslim University, Aligarh 202002, India
Amal S. Alali: Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
Elif Segah Oztas: Department of Mathematics, Kamil Ozdag Science Faculty, Karamanoglu Mehmetbey University, Karaman 70100, Türkiye
Pushpendra Sharma: Department of Mathematics, Faculty of Science, Aligarh Muslim University, Aligarh 202002, India

Mathematics, 2023, vol. 11, issue 6, 1-16

Abstract: Let k , m be positive integers and F 2 m be a finite field of order 2 m of characteristic 2. The primary goal of this paper is to study the structural properties of cyclic codes over the ring S k = F 2 m [ v 1 , v 2 , … , v k ] ⟨ v i 2 − α i v i , v i v j − v j v i ⟩ , for i , j = 1 , 2 , 3 , … , k , where α i is the non-zero element of F 2 m . As an application, we obtain better quantum error correcting codes over the ring S 1 (for k = 1 ). Moreover, we acquire optimal linear codes with the help of the Gray image of cyclic codes. Finally, we present methods for reversible DNA codes.

Keywords: Kronecker product; cyclic codes; Gray map; quantum codes; DNA codes (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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