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Maximal entanglement-assisted quantum error correction codes from the skew group ring \({\mathbb {F}}_4 \rtimes _{\varphi } G\) by a heuristic search scheme

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Abstract

Construction of maximal entanglement-assisted quantum error correction (EAQEC) codes is one of the fundamental problems of quantum computing and quantum information. The objective of this paper is twofold: firstly, to obtain all possible construction matrices of the linear codes over the skew group ring \({\mathbb {F}}_4 \rtimes _{\varphi } G,\) where G is the cyclic and dihedral groups of finite orders; and secondly, to obtain some good maximal EAQEC codes over the finite field \({\mathbb {F}}_4\) by using skew construction matrices. Additionally, to speed up the computational search time, we employ a nature inspired heuristic optimisation algorithm, the virus optimisation (VO) algorithm. With our method, we obtain a number of good maximal EAQEC codes over the finite field \({\mathbb {F}}_4\) in a reasonably short time. In particular, we improve the lower bounds of 18 maximal EAQEC codes that exist in the literature. Moreover, some of our EAQEC codes turn out to be also maximum distance separable (MDS) codes. Also, by using our construction matrices, we provide counterexamples to Theorems 4 and 5 of Lai et al. (Quantum Inf Process 13(4):957–990, 2014), on the non-existence of maximal EAQEC codes with parameters \([[n, 1, n; n-1]]\) and \([[n, n-1, 2;1]]\) for an even length n. We also give a counterexample to another Theorem found in Lai and Ashikhmin (IEEE Trans Inf Theory 64:(1), 622–639, 2018), which states that there is no entanglement-assisted stabilizer code with parameters \([[4, 2, 3;2]]_4.\)

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References

  1. Ajitha Shenoy, K.B., Biswas, S., Kurur, P.P.: Efficacy of the metropolis algorithm for the minimum-weight codeword problem using codeword and generator search spaces. IEEE Trans. Evolut. Comput. 24(4), 664–678 (2020)

    Article  Google Scholar 

  2. Aydin, N., Abualrub, T.: Optimal quantum codes from additive skew cyclic codes. Discrete Math. Algorithms Appl. 8(3), 1650037 (2016)

    Article  MathSciNet  Google Scholar 

  3. Aydin, N.: Some new linear codes from skew cyclic codes and computer algebra challenges. AAECC 30, 185–191 (2019)

    Article  MathSciNet  Google Scholar 

  4. Bag, T., Dinh, H., Upadhyay, Q., Ashish, K., Ramakrishna, B., Yamaka, W.: Quantum codes from skew constacyclic codes over the ring \({\mathbb{F}}_q[u, v]/\langle u^2 -1, v^2- 1, uv- vu \rangle \). Discrete Math. 343(3), 111737 (2020)

    Article  MathSciNet  Google Scholar 

  5. Tushar, B., Mohammad, A., Ghulam, M., Ashish, U.K.: Quantum codes from \((1-2u_1- 2u_2 -\dots -2u_m)\)-skew constacyclic codes over the ring \({\mathbb{F}}_q+u_1{\mathbb{F}}_q+\dots +u_{2m}{\mathbb{F}}_q\). Quantum Inf. Process. 18(9), 270 (2019)

    Article  ADS  Google Scholar 

  6. Bland, J.A., Baylis, A.T.: A tabu search approach to the minimum distance of error-correcting codes. Int. J. Electron. 79(6), 829–837 (1995)

    Article  Google Scholar 

  7. Bland, J.A.: Local search optimisation applied to the minimum distance problem. Adv. Eng. Inf. 21, 391–397 (2007)

    Article  Google Scholar 

  8. Bouzkraoui, H., Azouaoui, A., Hadi, Y.: New ant colony optimization for searching the minimum distance for linear codes. In: Advanced Communication Technologies and Networking (CommNet), International Conference on. IEEE (2018)

  9. Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24, 235–265 (1997)

    Article  MathSciNet  Google Scholar 

  10. Brun, T.A., Devetak, I., Hsieh, M.H.: Correcting quantum errors with entanglement. Science 314, 436–439 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  11. Bowen, G.: Entanglement required in achieving entanglement-assisted channel capacities. Phys. Rev. A 66, 052313 (2002)

    Article  ADS  Google Scholar 

  12. Calderbank, A., Rains, E.M., Shor, P.W., Sloane, N.J.A.: Quantum error correction via codes over GF(4). IEEE Trans. Inf. Theory 44(4), 1369–1387 (1998)

    Article  MathSciNet  Google Scholar 

  13. Carbas, S., Toktas, A., Ustun, D.: Nature-Inspired Metaheuristic Algorithms for Engineering Optimization Applications, vol. 1. Springer, Singapore (2021). ISBN:978-981-336-772-2

  14. Chen, Z., Zhou, K., Liao, Q.: Quantum identity authentication scheme of vehicular ad-hoc networks. Int. J. Theor. Phys. 58(1), 40–57 (2019)

    Article  Google Scholar 

  15. Dinh, H.Q., Bag, T., Upadhyay, A.K., Bandi, R., Tansuchat, R.: A class of skew cyclic codes and application in quantum codes construction. Discrete Math. 344(2), 112189 (2021)

    Article  MathSciNet  Google Scholar 

  16. Dougherty, S.T., Gildea, J., Taylor, R., Tylshchak, A.: Group rings, \(G\)-: codes and constructions of self-dual and formally self-dual codes. Designs, Codes Cryptogr. 86(9), 2115–2138 (2018)

    Article  MathSciNet  Google Scholar 

  17. Dougherty, S.T., Korban, A., Sahinkaya, S., Ustun, D.: Group matrix ring codes and constructions of self-dual codes. AAECC. https://doi.org/10.1007/s00200-021-00504-9

  18. Dougherty, S.T., Korban, A., Sahinkaya, S., Ustun, D.: Additive skew \(G\)-codes over finite fields. AAECC. https://doi.org/10.1007/s00200-021-00515-6

  19. Dougherty, S.T., Sahinkaya, S., Yildiz, B.: Skew \(G\)-codes. J. Algebra Appl. https://doi.org/10.1142/S0219498823500561

  20. Ezerman, M.F., Ling, S., Sole, P., Yemen, O.: From skew-cyclic codes to asymmetric quantum codes. Adv. Math. Commun. 5(1), 41–57 (2011)

    Article  MathSciNet  Google Scholar 

  21. Guenda, K., Jitman, S., Gulliver, T.A.: Constructions of good entanglement assisted quantum error correcting codes. Des. Codes Cryptogr. 86, 121–136 (2018)

    Article  MathSciNet  Google Scholar 

  22. Hurley, T.: Group rings and rings of matrices. Int. J. Pure Appl. Math. 31(3), 319–335 (2006)

    MathSciNet  MATH  Google Scholar 

  23. Korban, A., Sahinkaya, S., Ustun, D.: Generator matrices of maximal entanglement-assited quantum error correction codes from the skew group ring \({\mathbb{F}}_4 {\varphi } G\). https://sites.google.com/view/serap-sahinkaya/generator-matrices

  24. Lai, C.Y., Brun, T.A., Wilde, M.: Duality in entanglement-assisted quantum error correction. IEEE Trans. Inf. Theory 59(6), 4020–4024 (2013)

    Article  MathSciNet  Google Scholar 

  25. Lai, C., Brun, T.A., Wilde, M.: Dualities and identities for entanglement-assisted quantum codes. Quantum Inf. Process. 13(4), 957–990 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  26. Lai, C.Y., Ashikhmin, A.: Linear programming bounds for entanglement-assisted quantum error correcting codes by split weight enumerators. IEEE Trans. Inf. Theory 64(1), 622–639 (2018)

    Article  MathSciNet  Google Scholar 

  27. Lai, C.Y., Brun, T.A.: Entanglement-assisted quantum error correcting codes with imperfect ebits. Phys. Rev. A 86, 032319 (2012)

    Article  ADS  Google Scholar 

  28. Lai, C.Y., Brun, T.A.: Entanglement increases the error-correcting ability of quantum error-correcting codes. Phys. Rev. A 88, 012320 (2013)

    Article  ADS  Google Scholar 

  29. Li, J., Gao, J., Fu, F.W., Ma, F.: \({\mathbb{F} }_qR\)-linear skew constacyclic codes and their application of constructing quantum codes. Quantum Inf. Process. 19(7), 1–23 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  30. Liu, H., Liu, X.: New EAQECC codes from cyclic codes over\({\mathbb{F}}_q+u{\mathbb{F}}_q\). Quantum Inf. Process. 19(3), 1–6 (2020)

    MathSciNet  Google Scholar 

  31. Liu, X., Yu, L., Hu, P.: New entanglement-assisted quantum codes from k-Galois dual codes. Finite Fields Appl. 55, 21–32 (2019)

    Article  MathSciNet  Google Scholar 

  32. Lu, L., Li, R., Guo, L., Fu, Q.: Maximal entanglement-assisted quantum codes constructed from linear codes. Quantum Inf. Process. 14(1), 165–182 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  33. Lu, L., Li, R., Ma, W., Ma, Y., Liu, Y., Cao, H.: Entanglement-assisted quantum MDS codes from constacyclic codes with large minimum distance. Finite Fields Appl. 53, 309–325 (2018)

    Article  MathSciNet  Google Scholar 

  34. Lv, J., Li, R., Yao, Y.: Extended quasi-cyclic constructions of quantum codes and entanglement-assisted quantum codes. Comput. Appl. Math. 40(8), 1–20 (2021)

    Article  MathSciNet  Google Scholar 

  35. Qian, J., Zhang, L.: On MDS linear complementary dual codes and entanglement-assisted quantum codes. Des. Codes Cryptogr. 86(7), 1565–1572 (2018)

    Article  MathSciNet  Google Scholar 

  36. Qian, J., Zhang, L.: Entanglement-assisted quantum codes from arbitrary binary linear codes. Des. Codes Cryptogr. 77(1), 193–202 (2015)

    Article  MathSciNet  Google Scholar 

  37. Shor, P.W.: Scheme for reducing decoherence in quantum memory. Phys. Rev. A, Gen. Phys. 52(4), 2493–2496 (1995)

    Article  ADS  Google Scholar 

  38. Wang, J., Li, R., Lv, J., Guo, G., Liu, Y.: Entanglement-assisted quantum error correction codes with length \(n = q2 + 1\). Quantum Inf. Process. 18, 1–21 (2019)

    Article  Google Scholar 

  39. Wilde, M.M., Hsieh, M.H., Babar, Z.: Entanglement-assisted quantum turbo codes. IEEE Trans. Inf. Theory 60, 1203–1222 (2014)

    Article  MathSciNet  Google Scholar 

  40. Xiao, H., Zhang, Z.: Subcarrier multiplexing multiple-input multiple-output quantum key distribution with orthogonal quantum states. Quantum Inf. Process. 16(13), 1–18 (2017)

    ADS  MATH  Google Scholar 

  41. Xin, X., He, Q., Wang, Z., Yang, Q., Li, F.: Efficient arbitrated quantum signature scheme without entangled states. Mod. Phys. Lett. A 34(21), 1950166 (2019)

  42. Yao, Y., Ma, Y., Lv, J.: Quantum codes and entanglement-assisted quantum codes derived from one-generator quasi-twisted codes. Int. J. Theoret. Phys. 60(3), 1077–1089 (2021)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We thank the anonymous referees for their valuable comments and suggestions that helped to improve significantly the quality of this paper. This research work is part of the Scientific Research Project of Tarsus University with Project Number MF.20.003. The authors thank Tarsus University for providing workstations with high computation performance and providing the software package MAGMA, which was used for the computational part of this paper.

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Author notes

  1. Adrian Korban and Deniz Ustun contributed equally to this work.

Authors and Affiliations

  1. Department of Natural and Mathematical Sciences, Tarsus University, Mersin, 33400, Turkey

    Serap Şahinkaya

  2. Department of Mathematical and Physical Sciences, University of Chester, Thornton Science Park, Chester, CH2 4NU, UK

    Adrian Korban

  3. Department of Computer Engineering, Tarsus University, Mersin, 33400, Turkey

    Deniz Ustun

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  1. Serap Şahinkaya
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  2. Adrian Korban
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  3. Deniz Ustun
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Correspondence to Serap Şahinkaya.

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Şahinkaya, S., Korban, A. & Ustun, D. Maximal entanglement-assisted quantum error correction codes from the skew group ring \({\mathbb {F}}_4 \rtimes _{\varphi } G\) by a heuristic search scheme. Quantum Inf Process 21, 156 (2022). https://doi.org/10.1007/s11128-022-03500-1

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