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Hermitian dual-containing constacyclic codes over \(\mathbb {F}_{q^{2}}+{v_{1}}\mathbb {F}_{q^{2}}+\cdots +{v_{r}}\mathbb {F}_{q^{2}}\) and new quantum codes

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Abstract

In this paper, we study Hermitian dual-containing constacyclic codes over the finite ring \(R_{r}=\mathbb {F}_{q^{2}}+{v_{1}}\mathbb {F}_{q^{2}}+\cdots +{v_{r}}\mathbb {F}_{q^{2}}\), where q is a prime power and \({v_{i}}^{2}={v_{i}},v_{i}v_{j}=v_{j}v_{i}=0\) for 1 ≤ i,j ≤ r,i≠j. A necessary and sufficient condition is provided to determine whether a constacyclic code C over Rr is Hermitian dual-containing. Moreover, we propose a generalized Gray map ΦM to preserve the property of Hermitian dual-containing. Compared with some existing Gray maps, ΦM increases the possibility of making the minimum distance of ΦM(C) larger. As an application, some new quantum codes over \(\mathbb {F}_{q}\) are constructed from constacyclic codes over Rr.

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References

  1. Ashikhmin, A., Knill, E.: Nonbinary quantum stabilizer codes. IEEE Trans. Inform. Theory 47(7), 3065–3072 (2001)

    Article  MATH  Google Scholar 

  2. Ashraf, M., Mohammad, G.: Quantum codes from cyclic codes over \(\mathbb {F}_{3}+v\mathbb {F}_{3}\). Int. J. Quantum Inf. 12(6), 1450042 (2014)

    Article  MATH  Google Scholar 

  3. Ashraf, M., Mohammad, G.: Construction of quantum codes from cyclic codes over \(\mathbb {F}_{p}+v\mathbb {F}_{p}\). Int. J. Inf. Coding Theory 3(2), 137–144 (2015)

    MATH  Google Scholar 

  4. Ashraf, M., Mohammad, G.: Quantum codes from cyclic codes over \(\mathbb {F}_{q}+u\mathbb {F}_{q}+v\mathbb {F}_{q}+uv\mathbb {F}_{q}\). Quantum Inf. Process. 15, 4089–4098 (2016)

    Article  MATH  Google Scholar 

  5. Ashraf, M., Mohammad, G.: Quantum codes from cyclic codes over \(\mathbb {F}_{p}[u,v]/\langle u^{2}-1,v^{3}-v,uv-vu\rangle\). Cryptogr. Commun. 11, 325–335 (2019)

    Article  MATH  Google Scholar 

  6. Bag, T., Ashraf, M., Mohammad, G., Upadhyay, A.K.: Quantum codes from (1 − 2u1 − 2u2 −⋯ − 2um)-skew constacyclic codes over the ring Fq + u1Fq + ⋯ +u2mFq. Quantum Inf. Process. 18, 270 (2019)

    Article  Google Scholar 

  7. Blackmore, T., Norton, G.H.: Matrix-product codes over \(\mathbb {F}_{q}\). Appl. Algebra Eng. Commun. Comput. 12(6), 477–500 (2001)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  9. Cao, M., Cui, J.: Construction of new quantum codes via Hermitian dual-containing matrix-product codes. Quantum Inf. Process. 19(12), 1–26 (2020)

    Article  Google Scholar 

  10. Chen, B., Ling, S., Zhang, G.: Application of constacyclic codes to quantum MDS codes. IEEE Trans. Inform. Theory 61(3), 1474–1484 (2015)

    Article  MATH  Google Scholar 

  11. Dertli, A., Cengellenmis, Y., Eren, S.: On quantum codes obtained from cyclic codes over a2. Int. J. Quantum Inf. 13(3), 1550031 (2015)

    Article  MATH  Google Scholar 

  12. Dertli, A., Cengellenmis, Y., Eren, S.: Some results on the linear codes over the finite ring F2 + v1F2 + ⋯ + vrF2. Int. J. Quantum Inf. 14(1), 1650012 (2016)

    Article  MATH  Google Scholar 

  13. Diao, L., Gao, J., Lu, J.: Some results on \(\mathbb {Z}_{p}\mathbb {Z}_{p}[v]\)-additive cyclic codes. Adv. Math. Commun. 14(4), 555–572 (2020)

    Article  MATH  Google Scholar 

  14. Dinh, H.Q., Bag, T., Pathak, S., Upadhyay, A.K., Chinnakum, W.: Quantum codes obtained from constacyclic codes over a family of finite rings \(\mathbb {F}_{p}[u_{1},u_{2},\cdots ,u_{s}]\). IEEE Access 8, 194082–194091 (2020)

    Article  Google Scholar 

  15. Edel, Y.: Some good quantum twisted codes. https://www.mathi.uni-heidelberg.de/yves/Matritzen/QTBCH/QTBCHIndex.html

  16. Galindo, C., Hernando, F., Ruano, D.: New quantum codes from evaluation and matrix-product codes. Finite Fields Appl. 36, 98–120 (2015)

    Article  MATH  Google Scholar 

  17. Gao, J.: Quantum codes from cyclic codes over \(\mathbb {F}_{q}+v\mathbb {F}_{q}+v^{2}\mathbb {F}_{q}+v^{3}\mathbb {F}_{q}\). Int. J. Quantum Inf. 13(8), 1550063 (2015)

    Article  Google Scholar 

  18. Gao, J., Wang, Y.: Quantum codes derived from negacyclic codes. Int. J. Theor. Phys. 57, 682–686 (2018)

    Article  MATH  Google Scholar 

  19. Gao, J., Wang, Y.: u-Constacyclic codes over \(\mathbb {F}_{p}+u\mathbb {F}_{p}\) and their applications of constructing new non-binary quantum codes. Quantum Inf. Process. 17, 4 (2018)

    Article  MATH  Google Scholar 

  20. Gao, Y., Gao, J., Fu, F.: Quantum codes from cyclic codes over the ring \(\mathbb {F}_{q}+v_{1}\mathbb {F}_{q}+\cdots +v_{r}\mathbb {F}_{q}\). Appl. Algebra Eng. Commun. Comput. 30(2), 161–174 (2019)

    Article  Google Scholar 

  21. Islam, H., Patel, S., Prakash, O., Solé, P.: A family of constacyclic codes over a class of non-chain rings \(\mathcal {A}_{q,r}\) and new quantum codes. J. Appl. Math Comput. https://doi.org/10.1007/s12190-021-01623-9 (2021)

  22. Islam, H., Prakash, O.: New quantum codes from constacyclic and additive constacyclic codes. Quantum Inf. Process. 19, 319 (2020)

    Article  Google Scholar 

  23. Islam, H., Prakash, O., Bhunia, D.K.: Quantum codes obtained from constacyclic codes. Int. J. Theor. Phys. 58, 3945–3951 (2019)

    Article  MATH  Google Scholar 

  24. Kai, X., Zhu, S.: New quantum MDS codes from negacyclic codes. IEEE Trans. Inform. Theory 59(2), 1193–1197 (2013)

    Article  MATH  Google Scholar 

  25. Kai, X., Zhu, S.: Quaternary construction of quantum codes from cyclic codes over \(\mathbb {F}_{4}+u\mathbb {F}_{4}\). Int. J. Quantum Inf. 9(2), 689–700 (2011)

    Article  MATH  Google Scholar 

  26. Kai, X., Zhu, S., Li, P.: Constacyclic codes and some new quantum MDS codes. IEEE Trans. Inform. Theory 60(4), 2080–2086 (2014)

    Article  MATH  Google Scholar 

  27. Ketkar, A., Klappenecker, A., Kumar, S., Sarvepalli, P.: K: Nonbinary stabilizer codes over finite fields. IEEE Trans. Inform. Theory 52(11), 4892–4914 (2006)

    Article  MATH  Google Scholar 

  28. La Guardia, G.G.: Quantum codes derived from cyclic codes. Int. J. Theor. Phys. 56(8), 2479–2484 (2017)

    Article  MATH  Google Scholar 

  29. Li, J., Gao, J., Fu, F., Ma, F.: \(\mathbb {F}_{q}R\)-linear skew constacyclic codes and their application of constructing quantum codes. Quantum Inf. Process. 19, 193 (2020)

    Article  Google Scholar 

  30. Liu, X., Dinh, H. Q., Liu, H., Yu, L.: On new quantum codes from matrix product codes. Cryptogr. Commun. 10(4), 579–589 (2018)

    Article  MATH  Google Scholar 

  31. Liu, X., Liu, H.: Quantum codes from linear codes over finite chain rings. Quantum Inf. Process. 16(10), 240 (2017)

    Article  MATH  Google Scholar 

  32. Ma, F., Gao, J., Fu, F.: Constacyclic codes over the ring \(\mathbb {F}_{q}+v\mathbb {F}_{q}+v^{2}\mathbb {F}_{q}\) and their applications of constructing new non-binary quantum codes. Quantum Inf. Process. 17, 122 (2018)

    Article  Google Scholar 

  33. Ma, F., Gao, J., Fu, F.: New non-binary quantum codes from constacyclic codes over \(\mathbb {F}_{q}[u,v]/\langle u^{2}-1,v^{2}-v,uv-vu\rangle\). Adv. Math. Commun. 13(3), 421–434 (2019)

    Article  MATH  Google Scholar 

  34. Qian, J.: Quantum codes from cyclic codes over \(\mathbb {F}_{2}+v\mathbb {F}_{2}\). J. Inform. Comput. Sci. 10, 1715–1722 (2013)

    Article  Google Scholar 

  35. Qian, J., Ma, W., Guo, W.: Quantum codes from cyclic codes over finite ring. Int. J. Quantum Inf. 7(6), 1277–1283 (2009)

    Article  MATH  Google Scholar 

  36. Sari, M., Siap, I.: On quantum codes from cyclic codes over a class of nonchain rings. Bull. Korean Math. Soc. 53(6), 1617–1628 (2016)

    Article  MATH  Google Scholar 

  37. Shi, X., Huang, X., Yue, Q.: Construction of new quantum codes derived from constacyclic codes over \(\mathbb {F}_{q^{2}}+u\mathbb {F}_{q^{2}}+\cdots +u^{r-1}\mathbb {F}_{q^{2}}\). Appl. Algebra Eng. Commun. Comput. https://doi.org/10.1007/s00200-020-00415-1(2020)

  38. Song, H., Li, R., Liu, Y., Guo, G.: New quantum codes from matrix-product codes over small fields. Quantum Inf. Process. 19(8), 1–22 (2020)

    Article  Google Scholar 

  39. Tang, Y., Yao, T., Sun, Z., Zhu, S., Kai, X.: Nonbinary quantum codes from constacyclic codes over polynomial residue rings. Quantum Inf. Process. 19(3), 84 (2020)

    Article  Google Scholar 

  40. Tang, Y., Zhu, S., Kai, X., Ding, J.: New quantum codes from dual-containing cyclic codes over finite rings. Quantum Inf. Process. 15(11), 4489–4500 (2016)

    Article  MATH  Google Scholar 

  41. Wang, Y., Kai, X., Sun, Z., Zhu, S.: Quantum codes from Hermitian dual-containing constacyclic codes over \(\mathbb {F}_{q^{2}}+v\mathbb {F}_{q^{2}}\). Quantum Inf. Process. 20(3), 122 (2021)

    Article  Google Scholar 

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant Nos. U21A20428, 12171134, 61972126, 62002093), the Key Program for Outstanding Young Talents in University of Anhui Province of China (Grant No. gxyqZD2021137), and the Talent Scientific Research Fund of Hefei University (Grant No.18-19RC61).

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Authors and Affiliations

  1. School of Artificial Intelligence and Big Data, Hefei University, Anhui, 230601, Hefei, People’s Republic of China

    Yu Wang

  2. School of Mathematics, Hefei University of Technology, Hefei, 230009, Anhui, People’s Republic of China

    Xiaoshan Kai, Zhonghua Sun & Shixin Zhu

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  1. Yu Wang
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  2. Xiaoshan Kai
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Correspondence to Yu Wang.

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Wang, Y., Kai, X., Sun, Z. et al. Hermitian dual-containing constacyclic codes over \(\mathbb {F}_{q^{2}}+{v_{1}}\mathbb {F}_{q^{2}}+\cdots +{v_{r}}\mathbb {F}_{q^{2}}\) and new quantum codes. Cryptogr. Commun. 15, 145–158 (2023). https://doi.org/10.1007/s12095-022-00593-4

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