Skip to main content
SpringerLink
Log in

On quantum and LCD codes from the cyclic codes over the ring \(\mathbb {F}_q[\textsf{u}, \textsf{v},\textsf{w}]/\langle \textsf{u}^3-\textsf{u},\textsf{v}^2-\textsf{v}, \textsf{w}^2-\textsf{w}, \textsf{u}\textsf{v}, \textsf{v}\textsf{u}, \textsf{u}\textsf{w}, \textsf{w}\textsf{u}, \textsf{v}\textsf{w}-\textsf{w}\textsf{v} \rangle \)

  • Original Research
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

For an odd prime p and a positive integer r, let \(q=p^r\). The objective of this article is to study cyclic codes over the ring \(\mathcal {S}=\mathbb {F}_q[\textsf{u}, \textsf{v},\textsf{w}]/\langle \textsf{u}^3-\textsf{u},\textsf{v}^2-\textsf{v}, \textsf{w}^2-\textsf{w}, \textsf{u}\textsf{v}, \textsf{v}\textsf{u}, \textsf{u}\textsf{w}, \textsf{w}\textsf{u}, \textsf{v}\textsf{w}-\textsf{w}\textsf{v} \rangle \) and to construct new and better quantum and LCD codes from them. We give the structure of cyclic codes over the ring \(\mathcal {S}\) and obtain dual-containing codes over \(\mathbb {F}_q\) as the Gray images of dual-containing cyclic codes over \(\mathcal {S}\). Using these dual-containing codes, we obtain quantum codes and determine their parameters using the decomposition of cyclic codes over the ring \(\mathcal {S}\). We provide many new and better-than-existing quantum codes. We also give a method to obtain linear complementary dual (LCD) codes over \(\mathcal {S}\) using the decomposition of cyclic codes over the ring \(\mathcal {S}\). We obtain some optimal and best-known linear codes as Gray images of LCD codes over \(\mathcal {S}\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alahmadi, A., Altassan, A., AlKenani, A., Calkavur, S., Shoaib, H., Solé, P.: A multisecret-sharing scheme based on LCD codes. Mathematics 8(2), 272 (2020)

    Google Scholar 

  2. Ali, S., Mohammad, G., Khan, N., Sharma, P.: On quantum and LCD codes over the ring \(\mathbb{F} _q+ v\mathbb{F} _q+ v^2\mathbb{F} _q\). Quantum Inf. Process. 21(9), 306 (2022)

    Google Scholar 

  3. Ashraf, M., Khan, N., Mohammad, G.: New quantum and LCD codes over the finite field of odd characteristic. Int. J. Theor. Phys. 60(6), 2322–2332 (2021)

    MathSciNet  Google Scholar 

  4. 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)

    MathSciNet  Google Scholar 

  5. 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)

    MathSciNet  Google Scholar 

  6. 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(10), 4089–4098 (2016)

    MathSciNet  Google Scholar 

  7. Ashraf, M., Mohammad, G.: Quantum codes over \(\mathbb{F} _p\) 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)

    MathSciNet  Google Scholar 

  8. Aydin, N., Liu, P., Yoshino, B.: A database of quantum codes. http://quantumcodes.info/. Accessed 28 Feb 2023

  9. Bag, T., Upadhyay, A.K., Ashraf, M., Mohammad, G.: Quantum code from cyclic code over the ring \(\mathbb{F} _p[u]/\langle u^3 - u \rangle \). Asian Eur. J. Math. 13(1), 2050008 (2020)

    Google Scholar 

  10. Bhowmick, S., Fotue-Tabue, A., Martinez-Moro, E., Bandi, R., Bagchi, S.: Do non-free LCD codes over finite commutative Frobenius rings exist? Des. Codes Cryptogr. 88(5), 825–840 (2020)

    MathSciNet  Google Scholar 

  11. Bierbrauer, J., Edel, Y.: Quantum twisted codes. J. Combin. Des. 8(3), 174–188 (2000)

    MathSciNet  Google Scholar 

  12. Bosma, W., Cannon, J.J., Fieker, C., Steel, A. (Eds.): Handbook of Magma Functions, Edition 2.26-4 (2021). http://magma.maths.usyd.edu.au/calc/

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

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  15. Carlet, C., Guilley, S.: Complementary dual codes for counter-measures to side-channel attacks. Adv. Math. Commun. 10(1), 131–150 (2016)

    MathSciNet  Google Scholar 

  16. Carlet, C., Mesnager, S., Tang, C., Qi, Y.: Euclidean and Hermitian LCD MDS codes. Des. Codes Cryptogr. 86(11), 2605–2618 (2018)

    MathSciNet  Google Scholar 

  17. Carlet, C., Mesnager, S., Tang, C., Qi, Y., Pellikaan, R.: Linear codes over \(\mathbb{F} _q\) are equivalent to LCD codes for \(q > 3\). IEEE Trans. Inform. Theory 64(4), 3010–3017 (2018)

    MathSciNet  Google Scholar 

  18. Carlet, C., Mesnager, S., Tang, C., Qi, Y.: On \(\sigma \)-LCD codes. IEEE Trans. Inform. Theory 65(3), 1694–1704 (2019)

    MathSciNet  Google Scholar 

  19. Dertli, A., Cengellenmis, Y., Eren, S.: On quantum codes obtained from cyclic codes over \(A_2\). Int. J. Quantum Inf. 13(3), 1550031 (2015)

    MathSciNet  Google Scholar 

  20. Dinh, H.Q., Bag, T., Upadhyay, A.K., Bandi, R., Chinnakum, W.: On the structure of cyclic codes over \(\mathbb{F} _qRS\) and applications in quantum and LCD codes constructions. IEEE Access 8, 18902–18914 (2020)

    Google Scholar 

  21. Dougherty, S.T., Kim, J.-L., Ozkaya, B., Sok, L., Solé, P.: The combinatorics of LCD codes: linear programming bound and orthogonal matrices. Int. J. Inf. Coding Theory 4(2–3), 116–128 (2017)

    MathSciNet  Google Scholar 

  22. Durgun, Y.: On LCD codes over finite chain rings. Bull. Korean Math. Soc. 57(1), 37–50 (2020)

    MathSciNet  Google Scholar 

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

  24. Esmaeili, M., Yari, S.: On complementary-dual quasi-cyclic codes. Finite Fields Appl. 15(3), 375–386 (2009)

    MathSciNet  Google Scholar 

  25. Galvez, L., Kim, J.-L., Lee, N., Roe, Y.-G., Won, B.-S.: Some bounds on binary LCD codes. Cryptogr. Commun. 10(4), 719–728 (2018)

    MathSciNet  Google Scholar 

  26. Gao, J.: Some results on linear codes over \(\mathbb{F} _p +u\mathbb{F} _p +u^2\mathbb{F} _p\). J. Appl. Math. Comput. 47(1–2), 473–485 (2015)

    MathSciNet  Google Scholar 

  27. Gao, Y., Gao, J., Fu, F.W.: Quantum codes from cyclic codes over the ring \(\mathbb{F} _q + v_1\mathbb{F}_q +...+ v_r\mathbb{F}_q\). Appl. Algebra Eng. Commun. Comput. 30, 161–174 (2019)

  28. Grassl, M., Beth, T., Roetteler, M.: On optimal quantum codes. Int. J. Quantum Inf. 2(1), 55–64 (2004)

    Google Scholar 

  29. Grassl, M.: Code tables: bounds on the parameters of various types of codes available at http://www.codetables.de/. Accessed 28 Feb 2023

  30. Hammons, A., Kumar, P.V., Calderbank, A.R., Sloane, N.J.A., Solé, P.: The \(\mathbb{Z} _4\)-linearity of Kerdock, Preparata, Goethals, and related codes. IEEE Trans. Inform. Theory 40(2), 301–319 (1994)

    MathSciNet  Google Scholar 

  31. Huffman, W.C., Kim, J.L., Solé, P.: Concise Encyclopedia of Coding Theory. Chapman and Hall/CRC (2021)

  32. Islam, H., Prakash, O.: New quantum and LCD codes over finite fields of even characteristic. Def. Sci. J. 71(5), 66 (2021)

    Google Scholar 

  33. Islam, H., Prakash, O.: Construction of LCD and new quantum codes from cyclic codes over a finite non-chain ring. Cryptogr. Commun. 14(1), 59–73 (2022)

    MathSciNet  Google Scholar 

  34. Islam, H., Prakash, O.: Quantum codes from the cyclic codes over \(\mathbb{F} _p[u, v, w]/\langle u^2 - 1, v^2 - 1, w^2 -1, uv - vu, vw - wv, uw - wu \rangle \). J. Appl. Math. Comput. 60, 625–635 (2019)

    MathSciNet  Google Scholar 

  35. Islam, H., Prakash, O., Verma, R.K.: Quantum codes from the cyclic codes over \(\mathbb{F} _p[v, w]/\langle v^2 - 1, w^2 - 1, vw - wv \rangle \). Springer Proc. Math. Stat. 307, 67–74 (2020)

    Google Scholar 

  36. 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, 689–700 (2011)

    MathSciNet  Google Scholar 

  37. Koroglu, M.E., Siap, I.: Quantum codes from a class of constacyclic codes over group algebras. Malay. J. Math. Sci. 11(2), 289–301 (2017)

    MathSciNet  Google Scholar 

  38. Li, C., Ding, C., Li, S.: LCD cyclic codes over finite fields. IEEE Trans. Inform. Theory 63(7), 4344–4356 (2017)

    MathSciNet  Google Scholar 

  39. Ling, S., Xing, C.: Coding Theory: A First Course. Cambridge University Press, Cambridge (2004)

    Google Scholar 

  40. Liu, X., Liu, H.: LCD codes over finite chain rings. Finite Fields Appl. 34, 1–19 (2015)

    MathSciNet  Google Scholar 

  41. Liu, X., Liu, H.: \(\sigma \)-LCD codes over finite chain rings. Des. Codes Cryptogr. 88(4), 727–746 (2020)

    MathSciNet  Google Scholar 

  42. MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amstedram (1977)

    Google Scholar 

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

    MathSciNet  Google Scholar 

  44. Massey, J.L.: Linear codes with complementary duals. Discrete Math. 106(107), 337–342 (1992)

    MathSciNet  Google Scholar 

  45. Massey, J.L.: Reversible codes. Inf. Control. 7(3), 369–380 (1964)

    MathSciNet  Google Scholar 

  46. Peterson, W.W., Brown, D.T.: Cyclic codes for error detection. Proc. IRE 49(1), 228–235 (1961)

    MathSciNet  Google Scholar 

  47. Prange, E.: Cyclic Error-Correcting Codes in Two Symbols. TN-57-103. AirForce Cambridge Research Labs, Bedford (1957)

  48. Prange, E.: Some Cyclic Error-Correcting Codes with Simple Decoding Algorithm. TN-58-156. AirForce Cambridge Research Labs, Bedford (1958)

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

    Google Scholar 

  50. Qian, J.: Quantum codes from cyclic codes over \(\mathbb{F} _2 + v\mathbb{F} _2\). J. Inf. Comput. Sci. 10, 1715–1722 (2013)

    Google Scholar 

  51. Qian, L., Shi, M., Solé, P.: On self-dual and LCD quasi-twisted codes of index two over a special chain ring. Cryptogr. Commun. 11(4), 717–734 (2019)

    MathSciNet  Google Scholar 

  52. Rehman, N.U., Azmi, M., Mohammad, G.: New quantum and LCD Codes from cyclic codes over a finite non-chain ring. Rep. Math. Phys. 91(2), 237–250 (2023)

    MathSciNet  Google Scholar 

  53. Sari, M., Siap, I.: Quantum codes over a class of finite chain ring. Quantum Inf. Comput. 16(1–2), 39–49 (2016)

    MathSciNet  Google Scholar 

  54. Sendrier, N.: Linear codes with complementary duals meet the Gilbert–Varshamov bound. Discrete Math. 285(1–3), 345–347 (2004)

    MathSciNet  Google Scholar 

  55. Shi, M., Huang, D., Sok, L., Solé, P.: Double circulant LCD codes over \(\mathbb{Z} _4\). Finite Fields Appl. 58, 133–144 (2019)

    MathSciNet  Google Scholar 

  56. Shi, M., Li, S., Kim, J.L., Solé, P.: LCD and ACD codes over a noncommutative non-unital ring with four elements. Cryptogr. Commun. 14, 627–640 (2022)

    MathSciNet  Google Scholar 

  57. Shi, M., Özbudak, F., Xu, L., Solé, P.: LCD codes from tridiagonal Toeplitz matrices. Finite Fields Appl. 75, 101892 (2021)

    MathSciNet  Google Scholar 

  58. Shi, M., Zhu, H., Qian, L., Sok, L., Solé, L.: On self-dual and LCD double circulant and double negacirculant codes over \(\mathbb{F} _q + u\mathbb{F} _q\). Cryptogr. Commun. 12(1), 53–70 (2020)

    MathSciNet  Google Scholar 

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

    Google Scholar 

  60. Shor, P.W.: Algorithms for quantum computation: discrete logarithms and factoring. In: Proceedings 35th Annual Symposium on Foundations of Computer Science, pp. 124–134. IEEE Computer Society Press

  61. Sloane, N., Thompson, J.: Cyclic self-dual codes. IEEE Trans. Inf. Theory 29(3), 364–366 (1983)

    MathSciNet  Google Scholar 

  62. Sok, L., Shi, M., Solé, P.: Construction of optimal LCD codes over large finite fields. Finite Fields Appl. 50, 138–153 (2018)

    MathSciNet  Google Scholar 

  63. Steane, A.M.: Simple quantum error-correcting codes. Phys. Rev. A 54, 4741–4751 (1996)

    MathSciNet  Google Scholar 

  64. Stein, W.A., et al.: Sage Mathematics Software (Version 9.2), The Sage Development Team (2021). http://www.sagemath.org

  65. Verma, R.K., Prakash, O., Singh, A., Islam, H.: New quantum codes from skew constacyclic codes. Adv. Math. Commun. 17(4), 66 (2023)

    MathSciNet  Google Scholar 

  66. Yang, X., Massey, J.L.: The condition for a cyclic code to have a complementary dual. Discrete Math. 126(1–3), 391–393 (1994)

    MathSciNet  Google Scholar 

  67. Zhu, S., Wang, Y., Shi, M.: Some results on cyclic codes over \(\mathbb{F} _2 + u\mathbb{F} _2\). IEEE Trans. Inform. Theory 56(4), 1680–1684 (2010)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are thankful to anonymous reviewers for their valuable comments and suggestions that contributed to improving the quality of the paper.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, IIT (BHU), Varanasi, Uttar Pradesh, 221005, India

    Pradeep Rai & Ashok Ji Gupta

  2. CAIR, DRDO, C.V. Raman Nagar, Bengaluru, Karnataka, 560093, India

    Bhupendra Singh

Authors
  1. Pradeep Rai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bhupendra Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ashok Ji Gupta
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ashok Ji Gupta.

Ethics declarations

Competing Interests

The authors have no conflict of interest to declare that is relevant to the content of this paper.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rai, P., Singh, B. & Gupta, A.J. On quantum and LCD codes from the cyclic codes over the ring \(\mathbb {F}_q[\textsf{u}, \textsf{v},\textsf{w}]/\langle \textsf{u}^3-\textsf{u},\textsf{v}^2-\textsf{v}, \textsf{w}^2-\textsf{w}, \textsf{u}\textsf{v}, \textsf{v}\textsf{u}, \textsf{u}\textsf{w}, \textsf{w}\textsf{u}, \textsf{v}\textsf{w}-\textsf{w}\textsf{v} \rangle \). J. Appl. Math. Comput. 70, 1241–1262 (2024). https://doi.org/10.1007/s12190-024-02002-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-024-02002-w

Keywords

Mathematics Subject Classification

Search

Navigation