Improved and Novel Quantum Codes through CSS Construction from Cyclic Codes over Commutative Non-Local Ring  Fp×(Fp + αFp + α2 Fp)

Yadav, Arun Kumar; Pruthi, Manju

doi:10.47974/JDMSC-2001

Open Access Research Article

Improved and Novel Quantum Codes through CSS Construction from Cyclic Codes over Commutative Non-Local Ring Fp×(Fp + αFp + α2 Fp)

^*Arun Kumar YadavCorresponding authorarunyadav0307@gmail.comDepartment of MathematicsMeerpurIndira Gandhi UniversityRewari, Haryana, IndiaView full profile → , Manju Pruthimanju.pruthi@yahoo.comDepartment of MathematicsMeerpurIndira Gandhi UniversityRewari, Haryana, IndiaView full profile →

* Corresponding author · click or hover a name for details

pp. 1949–1961Vol. 27Issue 6September 2024DOI: 10.47974/JDMSC-2001 Crossmark XML

Received:: 08 Oct 2024
Published Online:: 16 Sep 2024
Article type:: Research Article
Language:: EN
Article no.:: JDMSC-2001
Pages:: 1949–1961

Abstract

In this study, by using CSS construction, we present the new quantum codes derived from cyclic codes over the commutative non - local ring R = Fp×(Fp + αFp + α2 Fp), where p is an odd prime, α3 = α. Firstly, we investigate the structure of the ring R and study linear codes by defining Lee weight over R. We define a distance – preserving Gray map from R→Fp4 and discuss the cyclic codes with their dual code by Euclidean inner products on R. Additionally, we find the quantum Singleton defect(QSD) of the quantum code, which is denoted as n + 2 – k – 2d. Magma Software will be employed to provide relevant illustrations for enhanced comprehension.

Keywords

Quantum codes Gray map Non-local ring Cyclic codes

Subject Classifications

94B0594B1594B3594B6011T7114H20

References

[1] Aly, Salah A., Andreas Klappenecker, and Pradeep Kiran Sarvepalli. “On quantum and classical BCH codes.” IEEE Transactions on Information Theory 53, no. 3, 1183-1188 (2007).
[2] Ashraf, Mohammad, and Ghulam Mohammad. “Quantum codes from cyclic codes over F3 + vF3.” International Journal of Quantum Information 12, no. 06, 1450042 (2014).
[3] Ashraf, Mohammad, and Ghulam Mohammad. “Construction of quantum codes from cyclic codes over Fp + vFp.” International Journal of Infor. and Coding Theory 3, no. 2, 137-144 (2015).
[4] Bag, Tushar, Ashish K. Upadhyay, Mohammad Ashraf, and Ghulam Mohammad. “Quantum codes from cyclic codes over the ring
Fp [u]/⟨u3 – u⟩.” Asian-European Journal of Mathematics 12, no. 07, 2050008 (2019).
[5] Bosma, Wieb, John Cannon, and Catherine Playoust. “The Magma algebra system I: The user language.” Journal of Symbolic Computation 24, no. 3-4, 235-265 (1997).
[6] Çalışkan, Fatma, and Refia Aksoy. “Linear codes over F2 × (F2 + vF2) and the MacWilliams identities.” Applicable Algebra in Engineering, Communication and Computing 31, no. 2 : 135-147 (2020).
[7] Calıskana, Fatma, and Refia Aksoyb. “Cyclic codes over F2 × (F2 + vF2) and binary quantum codes.” Filomat 37, no. 15, 5137-5147 (2023).
[8] Çalışkan, Fatma, Tülay Yıldırım, and Refia Aksoy. “Non-Binary Quantum Codes from Cyclic Codes over Fp × (Fp + vFp).” International Journal of Theoretical Physics 62, no. 2, 29 (2023).
[9] Calderbank, A. Robert, Eric M. Rains, Peter M. Shor, and Neil JA Sloane. “Quantum error correction via codes over GF (4).” IEEE Transactions on Information Theory 44, no. 4, 1369-1387 (1998).
[10] Dinh, Hai Q. “Constacyclic codes of length ps over Fpm + uFpm.” Journal of Algebra 324, no. 5 940-950 (2010).
[11] Gao, Jian. “Quantum codes from cyclic codes over Fq + vFq + v2 Fq + v3 Fq.” International Journal of Quantum Information 13, no. 08, 1550063 (2015).
[12] Gao, Yun, Jian Gao, and Fang-Wei Fu. “Quantum codes from cyclic codes over the ring Fq + v1 Fq + v2 Fq + ... + vr Fq.” Applicable Algebra in Engineering, Communication and Computing 30, 161-174 (2019).
[13] Gao, Jian, and Yongkang Wang. “u-Constacyclic codes over Fp + vFp and their applications of constructing new non-binary quantum codes.” Quantum Information Processing 17, 1-9 (2018).
[14] Gao, Jian, and Yongkang Wang. “New non-binary quantum codes derived from a class of linear codes.” IEEE Access 7, 26418-26421 (2019).
[15] Gao, Jian. “Some results on linear codes over Fp + uFp + u2 Fp.” Journal of Applied Mathematics & Computing 47, no. 1-2, 473 (2015).
[16] Hammons, A. Roger, P. Vijay Kumar, A. Robert Calderbank, Neil JA Sloane, and Patrick Solé. “The Z4-linearity of Kerdock, Preparata, Goethals, and related codes.” IEEE Transactions on Information Theory 40, no. 2, 301-319 (1994).
[17] H. Islam, O. Prakash, and D. K. Bhunia, “Algebraic structure of cyclic codes over M3(Fp),” J. Discret. Math. Sci. Cryptogr., 27, no. 1, pp. 189–201 (2024).
[18] La Guardia, Giuliano G., and Reginaldo Palazzo Jr. “Constructions of new families of nonbinary CSS codes.” Discrete mathematics 310, no. 21, 2935-2945 (2010).
[19] Mohammed, Mohanad Ali, Ameer J. Munshid, and Mehdi Alaeiyan. “Cyclic codes of length pn over Zpm” Journal of Discrete Mathematical Sciences and Cryptography 24, no. 2, 579-588 (2021).
[20] Shor, Peter W. “Scheme for reducing decoherence in quantum computer memory.” Physical review A 52, no. 4, R2493 (1995).

Views: 210Downloads: 83Citations: 1