Quantum codes derived from constacyclic codes over Z3 [u,v]/

Materials and Methods

Preliminaries

The ring 𝑅 = ℤ 3 + 𝑢ℤ 3 + 𝑣ℤ 3 + 𝑢𝑣ℤ 3 , with 𝑢 2 = 1, 𝑣 2 = 1, and 𝑢𝑣 = 𝑣𝑢, is a principal ideal ring containing 3 4 elements, with characteristic 3. The maximal ideals of 𝑅 appeared in [18] which are 〈2 + 𝑢 + 2𝑣〉, 〈2 + 𝑢 + 𝑢𝑣〉, 〈𝑢 + 𝑣 + 2𝑢𝑣〉, 〈2𝑢 + 2𝑣 + 2𝑢𝑣〉.

Notable units of 𝑅 include 1 -𝑢 + 𝑣 -2𝑢𝑣, 1 + 𝑢 -𝑣 -2𝑢𝑣, 1 -𝑢 -𝑣 + 2𝑢𝑣, 1 + 2𝑢 -2𝑣 -2𝑢𝑣, 1 -2𝑢 + 2𝑣 -2𝑢𝑣, and 1 -2𝑢 -2𝑣 + 2𝑢𝑣.

We take 𝜚 as a unit of 𝑅 for simplicity's sake, and observe that 𝜚 -1 = 𝜚 in every instance. Let's define 𝑏 1 = 1 + 𝑢 + 𝑣 + 𝑢𝑣, 𝑏 2 = 1 -𝑢 + 𝑣 -𝑢𝑣, 𝑏 3 = 1 + 𝑢 -𝑣 -𝑢𝑣, 𝑏 4 = 1 -𝑢 -𝑣 + 𝑢𝑣.

It follows that 𝑏 𝑖 2 = 𝑏 𝑖 , 𝑏 𝑖 𝑏 𝑗 = 0, and ∑ 4 𝑖=1 𝑏 𝑖 = 1 ; 𝑖, 𝑗 = 1,2,3,4 with 𝑖 ≠ 𝑗.

Utilizing the Chinese Remainder Theorem, a simple method to express the ring is as:

Consequently, an arbitrary element 𝑒 = 𝑒 1 + 𝑢𝑒 2 + 𝑣𝑒 3 + 𝑢𝑣𝑒 4 of 𝑅, where 𝑒 𝑖 ∈ ℤ 3 , has a unique representation 𝑒 = ∑ 4 𝑖=1 𝑏 𝑖 𝑘 𝑖 , with 𝑘 𝑖 ∈ 𝑍 3 for 1 ≤ 𝑖 ≤ 4. We commence our conversation by defining a few key terms:

1. The number of places at which two vectors differ, represented by d(x,y), is the Hamming distance between 𝐱 = (𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 ) and 𝐲 = (𝑦 1 , 𝑦 2 , … , 𝑦 𝑛 ).

2. The number of nonzero 𝑥 𝑖 in a vector 𝐱 = (𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 ) is its Hamming weight, represented by 𝑤𝑡(𝐱).

3. 𝐱 ⋅ 𝐲 = 𝑥 0 𝑦 0 + 𝑥 Then R is 𝒞, 𝒩 and 𝜚-CC codes of length n if ∁(𝐶) = 𝐶, ℵ(𝐶) = 𝐶 and 𝜏 𝜚 (𝐶) = 𝐶 respectively.

7. Let 𝑝(𝑥) = 𝑎 0 + 𝑎 1 𝑥 + ⋯ + 𝑎 𝑛 𝑥 𝑛 be a polynomial with coefficients from an arbitrary field. The polynomial is termed reciprocal or reflected (denoted 𝑝 * or 𝑝 𝑅 ) if 𝑝 * (𝑥) = 𝑎 𝑛 + 𝑎 𝑛-1 𝑥 + ⋯ + 𝑎 0 𝑥 𝑛 = 𝑥 𝑛 𝑝(𝑥 -1 ).

8. A polynomial 𝑓(𝑥) is defined as self-reciprocal if 𝑝(𝑥) = 𝑝 * (𝑥).

Gray map on R

The gray map Ψ: 𝑅 → ℤ 3 4 is defined as

Since Ψ is a linear map from (𝑅 𝑛 , 𝑑 𝐿 ) to (ℤ 3 𝑛 , 𝑑 𝐻 ) and preserves distances as an isometric mapping.

where k is the dimension of the linear code ψ(𝒫) and d L is minimum Lee distance of the linear code 𝒫.

Theorem 2.2.2 [18]: If 𝒫 is a linear code, then Ψ(𝒫 ⊥ ) = (Ψ(𝒫)) ⊥ .

Quantum codes from 𝝔-CC Codes

For 𝑖 = 1,2,3,4, let 𝑆 𝑖 be linear codes of length 𝑛 over ℤ 3 . For a linear code 𝒫, we define

Clearly, 𝒫 1 , 𝒫 2 , 𝒫 3 , and 𝒫 4 are linear codes over ℤ 3 of length 𝑛.

We explore the connections among CC codes, 𝒩 codes, and 𝒞 codes by analyzing various units.

Theorem 2.3.3 For ϱ = 1 + u -v -2uv, the code 𝒫 is a ϱ-CC code if and only if 𝒫 1 , 𝒫 3 , and 𝒫 4 are 𝒩 codes and 𝒫 2 is a 𝒞 code.

Proof. For any 𝜉 = (𝜉 0 , 𝜉 1 , … , 𝜉 𝑛-1 ) ∈ 𝑅 𝑛 ,

Where 𝜉 𝑗 = 𝜛 1 𝜏 + 𝜛 2 𝜂 + 𝜛 3 𝛾 + 𝜛 4 𝛿, with 𝜏 = (𝜏 0 , 𝜏 1 , … , 𝜏 𝑛-1 ) ∈ 𝒫 1 , 𝜂 = (𝜂 0 , 𝜂 1 , … , 𝜂 𝑛-1 ) ∈ 𝒫 2 , 𝛾 = (𝛾 0 , 𝛾 1 , … , 𝛾 𝑛-1 ) ∈ 𝒫 3 , and 𝛿 = (𝛿 0 , 𝛿 1 , … , 𝛿 𝑛-1 ) ∈ 𝒫 4 , for all 𝜏 𝑗 , 𝜂 𝑗 , 𝛾 𝑗 , 𝛿 𝑗 ∈ ℤ 3 for 𝑗 = 0,1, … , 𝑛 -1.

This expression simplifies to demonstrate that 𝒫 1 , 𝒫 Thus,

Theorem 2.3.5 For ϱ = 1 -u -v + 2uv, the code 𝒫 is a ϱ-CC code if and only if 𝒫 2 , 𝒫 3 , and 𝒫 4 are 𝒩 codes and 𝒫 1 is a 𝒞 code.

Proof. For any 𝜉 = (𝜉 0 , 𝜉 1 , … , 𝜉 𝑛-1 ) ∈ 𝑅 𝑛 ,

And 𝛿 = (𝛿 0 , 𝛿 1 , … , 𝛿 𝑛-1 ) ∈ 𝒫 4 , for all 𝜏 𝑗 , 𝜂 𝑗 , 𝛾 𝑗 , 𝛿 𝑗 ∈ ℤ 3 for 𝑗 = 0,1, … , 𝑛 -1.

For the 𝜚-CC code 𝑅, 𝜏 𝜌 (𝜉) = ((1 -𝑢 -𝑣 + 2𝑢𝑣)𝜉 𝑛-1 , 𝜉 0 , 𝜉

Conclusion

In our study, we identified the units that are self-inverse and utilized them to analyze the QCs of the ring 𝑅 = ℤ 3 + 𝑢ℤ 3 + 𝑣ℤ 3 + 𝑢𝑣ℤ 3 , with 𝑢 2 = 1, 𝑣 2 = 1, and 𝑢𝑣 = 𝑣𝑢.

Open Problem

Future investigations could delve into additional QCs over 𝑍 3 by employing different gray maps on ℤ 3 + 𝑢ℤ 3 + 𝑣ℤ 3 + 𝑢𝑣ℤ 3 while adhering to the same constraints. Moreover, it would be beneficial to explore QCs in this ring under varying conditions and to compute QCs using alternative rings.

References

1. Alahmadi A, et al. New quantum codes from constacyclic codes over a non-chain ring. Quantum Inf Process. 2021;20(2):117. Available from: https://link.springer.com/article/10.1007/s11128-020-02977-y
2. Ashraf M, Mohammad G. Quantum codes from cyclic codes F₃+vF₃. Int J Quantum Inf. 2014;12(6):1450042.
3. Ashraf M, Mohammad M. Quantum codes over Fp from cyclic codes over Fp[u, v]/<u²-1, v³-v, uv-vu>. Cryptogr Commun. 2018;11(6):1-11. Available from: http://dx.doi.org/10.1142/S0219749914500427
4. Alkenani AN, Ashraf M, Mohammad G. Quantum codes from constacyclic codes over the ring Fq[u₁,u₂]/<u₁²=u₁, u₂²-u₂, u₁u₂=u₂u₁>. Mathematics. 2020;8(5):781. Available from: https://doi.org/10.3390/math8050781
5. Bag T, Dinh HQ, Upadhyay AK, Bandi R, Yamaka W. Quantum codes from skew constacyclic codes over the ring Fq[u,v]/<u²-1,v²-1,uv=vu>. Discrete Math. 2021;343(3):111737. Available from: http://dx.doi.org/10.1016/j.disc.2019.111737
6. Calderbank AR, Rains EM, Shor PM, Sloane NJA. Quantum error correction via codes over GF(4). IEEE Trans Inf Theory. 1998;44(4):1369-1387.
7. Dinh HQ, Bag T, Upadhyay AK, Ashraf M, Mohammad G, Chinnakum W. Quantum codes from a class of constacyclic codes over finite commutative rings. J Algebra Appl. 2020;19(12):2150003. Available from: http://dx.doi.org/10.1142/S0219498821500031
8. Dinh HQ, Kumar K, Singh AK. A study of quantum codes obtained from cyclic codes over a non-chain ring. Cryptogr Commun. 2022;14(4):909-923. Available from: https://doi.org/10.1007/s12095-022-00567-6
9. Gowdhaman K, Mohan C, Chinnapillai D, Gao J. Construction of quantum codes from λ-constacyclic codes over the ring Fp[u,v]/<u³=u,v³=v,uv=vu>. J Appl Math Comput. 2021;65(6):611-622. Available from: http://dx.doi.org/10.1007/s12190020-01407-7
10. Islam H, Prakash O. Quantum codes from the cyclic codes over Fp[u,v,w]/<u²-1, v²-1, w²-1, uv-vu, vw-wv, wu-uw>. J Appl Math Comput. 2019;60(1-2):625-635. Available from: http://dx.doi.org/10.1007/s12190-018-01230-1
11. Islam H, Verma RK, Prakash O. A family of constacyclic codes over F(pᵐ)[ν,ω]/<ν²-1, ω²-1, νω-ων>. Int J Inf Coding Theory. 2020;5(3-4):198-210. Available from: http://dx.doi.org/10.1504/IJICOT.2019.10026515
12. Li J, Gao J, Wang Y. Quantum codes from (1-2v)-constacyclic codes over the ring Fq+uFq+vFq+uvFq. Discrete Math Algorithms Appl. 2018;10(4):1850046. Available from: http://dx.doi.org/10.1142/S1793830918500465
13. Ma F, Gao J, Fu FW. Constacyclic codes over the ring Fp+vFp+v²Fp and their applications of constructing new non-binary quantum codes. Quantum Inf Process. 2018;17(6):1-19. Available from: https://doi.org/10.1007/s11128-018-1898-6
14. Ma F, Gao J, Fu FW. New non-binary quantum codes from constacyclic codes over Fq[u,v]/<u²-1,v²-v,uv-vu>. Adv Math Commun. 2019;13(3):421-434. Available from: https://doi.org/10.3934/amc.2019027
15. Ozen M, Ozzaim NT, Ince H. Quantum codes from cyclic codes over F₃+uF₃+vF₃+uvF₃. J Phys Conf Ser. 2016;766(1):012020. Available from: https://doi.org/10.1088/1742-6596/766/1/012020
16. Qian J. Quantum codes from cyclic codes over F₂+vF₂. J Inf Comput Sci. 2013;10:1715-1722. Available from: http://dx.doi.org/10.12733/jics20101705
17. Singh J, Mor P. Quantum codes obtained through constacyclic codes over Z₃+νZ₃+ωZ₃+νωZ₃. Eur J Pure Appl Math. 2021;14(3):1082-1097.
18. Shor PW. Scheme for reducing decoherence in quantum memory. Phys Rev A. 1995;52:2493-2496.
19. Singh AK, Pattanayek S, Kumar P. On quantum codes from cyclic codes over F₂+uF₂+u²F₂. Asian-Eur J Math. 2018;11(1):1850009. Available from: https://doi.org/10.1142/S1793557118500092
20. Pinki, Yadav S, Singh B, Mor P. Quantum codes obtained through (1-2vw)-constacyclic codes over Z₃+νZ₃+ωZ₃+νωZ₃. Adv Appl Discrete Math. 2023;38(1):1-14. Available from: https://doi.org/10.17654/0974165823015