Constacyclic Codes Over 𝑭𝒒[𝒖] /〈𝒖𝟑 = 𝟎〉 and Their Application of Constructing Quantum Codes

Yıl 2022, Cilt: 5 Sayı: 2, 12 - 22, 16.01.2023

Zineb Hebbache

https://doi.org/10.53508/ijiam.1127019

Öz

Let 𝑅 = 𝐹𝑞+u𝐹𝑞+𝑢^2𝐹𝑞, 𝑢^3=0 be a finite chain ring. In this paper, we give the structure of constacyclic codes over 𝑅 and obtain self-orthogonal codes over 𝐹𝑞 by using the Gray map from 𝑅𝑛 to 𝐹𝑞^(3𝑛). As an application, we present a construction of quantum codes from the codes obtained from this class.

Anahtar Kelimeler

Constacyclic codes, Gray map, Euclidean self-orthogonal, quantum codes.

Kaynakça

• [1] M. Ashraf and G. Mohammad, Quantum codes over 𝐹𝑝 from cyclic codes over 𝐹𝑝[𝑢, 𝑣]/ 〈𝑢^2 −1, 𝑣^2 − 1 〉 . Cryptogr. Commun. 11(2019), pp. 325--335.
• [2] A-R. Calderbank, E-M. Rains, P-M. Shor and N-J-A. Sloane, Quantum error correction via codes over GF(4) IEEE Trans. Inform. Theory, 44(1998), pp. 1369--1387.
• [3] Z. Chen, K. Zhou and Q. Liao, Quantum identity authentication scheme of vehicular adhoc networks, Int. J. Theor. Phys. ,58(2019), pp. 40--57.
• [4] J. Gao, Quantum codes from cyclic codes over 𝐹𝑞 + 𝑣𝐹𝑞 + 𝑣^2𝐹𝑞 + 𝑣^3𝐹𝑞. Int. J. Quantum Inf. 8(2015), pp. 1550063(1-8).
• [5] F. Ma, J. Gao and F-W. Fu, Constacyclic codes over the ring 𝐹𝑞 + 𝑣𝐹𝑞 + 𝑣2𝐹𝑞 and their applications of constructing new non-binary quantum codes, Quantum Inf. Process., 17, 122 (2018).
• [6] Y. Gao, J. Gao and F-W. Fu, On Quantum codes from cyclic codes over the ring 𝐹𝑞 + 𝑣𝐹𝑞 + ⋯ +𝑣𝑟𝐹 𝑞, Appl. Algebra Eng. Commun. Comput., 2(2019), pp. 161--174.
• [7] M. Guzeltepe and M. Sari, Quantum codes from codes over the ring 𝐹𝑞 + 𝛼𝐹𝑞 Quantum Inf. Process., 12(2019), 365.
• [8] F. Ma, J. Gao and F-W. Fu, New non-binary quantum codes from constacyclic codes𝐹𝑝 [𝑢, 𝑣]/ 〈𝑢^2 − 1, 𝑣^2 − 1 〉, Adv. Math. Commun. 2(2019), pp. 421--434.
• [9] J. Mi, X. Cao, S. Xu and G. Luo, Quantum codes from Hermitian dual-containing cyclic codes Int. J. Comput. Math., 3(2016).
• [10] Shor, P.: Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A 4(1995), 2493--2496.
• [11] M. Özen, N. Özzaim and H. Ince, Quantum codes from cyclic codes over 𝐹3 + 𝑢𝐹3 + 𝑣𝐹3 +𝑢𝑣𝐹3, Int. Conf. Quantum Sci. Appl. J. Phys. Conf. Ser. 766(2016), pp. 012020-1--012020-6.
• [12] H. Xiao, Z. Zhang and A. Chronopoulos, New construction of quantum error avoiding codes via group representation of quantum stabilizer, codes. Eur. Phys. J. C 77(2017), pp. 667--680.
• [13] H. Xiao and Z. Zhang, Subcarrier multiplexing multiple-input multiple-output quantum key distribution with orthogonal quantum states, Quantum Inf. Process., 16(2017), pp.1--18 .
• [14] X. Xin, Q. He, Z. Wang, Q. Yang and F. Li, Efficient arbitrated quantum signature scheme without entangled states, Mod. Phys. Lett. A 34(2019), 1950166.
• [15] J. Gao , F.W. Fu, L. Xiao and R.K. Bandi, Double cyclic codes 𝐹𝑞 + 𝑢𝐹𝑞 + 𝑢^2𝐹𝑞, Discrete Math. Algorithms Appl., 7(2015), pp. 1550058.
• [16] W-C. Huffman and V. Pless, Fundamentals of Error Correcting Codes, The United states of America by Combridge. University Press, New york, 2003.
• [17] Zheng, X., Bo, K.: Cyclic codes and 𝜆_{1} + 𝜆_{2}𝑢 + 𝜆_{3}𝑣 + 𝜆_{4}𝑢𝑣 −constacyclic codes over 𝐹𝑝 +𝑢𝐹𝑝 + 𝑣𝐹𝑝 + 𝑢𝑣𝐹𝑝. Appl. Math. Comput. 306(2017), pp. 86-91 .