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Construction of LCD and new quantum codes from cyclic codes over a finite non-chain ring

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Abstract

For an odd prime p and q = pr, this paper deals with LCD codes obtained from cyclic codes of length n over a finite commutative non-chain ring \(\mathcal {R}=\mathbb {F}_{q}[u,v]/\langle u^{2}-\alpha u,v^{2}-1, uv-vu\rangle \) where α is a non-zero element in \(\mathbb {F}_{q}\). Initially, we impose certain conditions on the generator polynomials of cyclic codes when \(\gcd (n,p)=1\) and \(\gcd (n,p)\neq 1\), respectively so that these codes become LCD. Then, by defining a Gray map ψ, we show that the Gray image of an LCD code of length n over \(\mathcal {R}\) is an LCD code of length 4n over \(\mathbb {F}_{q}\). In this way, we obtain many optimal and best-known linear codes (BKLC) from the Gray images of both cyclic and LCD codes over \(\mathcal {R}\). Eventually, by applying the CSS construction on cyclic codes over \(\mathcal {R}\) that contain their Euclidean duals, we determine many superior quantum codes compared to the existing codes in the recent references.

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References

  1. Abualrub, T., Siap, I.: Cyclic codes over the rings \(\mathbb {Z}_{2}+u\mathbb {Z}_{2}\) and \(\mathbb {Z}_{2}+u\mathbb {Z}_{2}+u^{2}\mathbb {Z}_{2}\). Des. Codes Cryptogr. 42(3), 273–287 (2007)

    Article  MathSciNet  Google Scholar 

  2. Abualrub, T., Ghrayeb, A., Oehmke, R.: A mass formula and rank of \(\mathbb {Z}_{4}\) cyclic codes of Length 2e. IEEE Trans. Inform. Theory 50(12), 3306–3312 (2004)

    Article  MathSciNet  Google Scholar 

  3. Alahmadi, A., Islam, H., Prakash, O., Solé, P., Alkenani, A., Muthana, N., Hijazi, R.: New quantum codes from constacyclic codes over a non-chain ring. Quantum Inf. Process. 20(2), Paper number: 60 17pp (2021)

  4. 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(2), 325–335 (2019)

    Article  MathSciNet  Google Scholar 

  5. 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)

    Article  MathSciNet  Google Scholar 

  6. Bosma, W., Cannon, J.: Handbook of magma functions. University of sydney (1995)

  7. Bonnecaze, A., Udaya, P.: Cyclic codes and self-dual codes over \(\mathbb {F}_{2}+u\mathbb {F}_{2}\). IEEE Trans. Inform. Theory 45(4), 1250–1255 (1999)

    Article  MathSciNet  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  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. 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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  14. 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  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

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

    Article  MathSciNet  Google Scholar 

  18. 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)

    Article  MathSciNet  Google Scholar 

  19. 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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  22. Grassl, M.: Code Tables: Bounds on the parameters of various types of codes available at http://www.codetables.de/ accessed on 20/04/2021

  23. 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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  25. Islam, H., Prakash, O., Bhunia, D.K.: Quantum codes obtained from constacyclic codes. Internat. J. Theoret. Phys. 58(11), 3945–3951 (2019)

    Article  MathSciNet  Google Scholar 

  26. 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)

    MATH  Google Scholar 

  27. Islam, H., Prakash, O., Verma, R.K.: A family of constacyclic codes over \(\mathbb {F}_{p^{m}}[v,w]/\langle v^{2}-1,w^{2}-1,vw-wv\rangle \). Int. J. Inf. Coding Theory 5(3/4), 198–210 (2020)

    Google Scholar 

  28. Islam, H., Prakash, O., Verma, R.K.: New quantum codes from constacyclic codes over the ring Rk,m. Adv. Math. Commun. https://doi.org/10.3934/amc.2020097 (2020)

  29. Islam, H.: Prakash: New quantum codes from constacyclic and additive constacyclic codes. Quantum Inf. Process. 19(9), Paper number: 319 17pp (2020)

  30. 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  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

  36. 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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  38. 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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  40. 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)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  42. Shor, P.W.: Algorithms for quantum computation: discrete logarithms and factoring. Proceedings 35th Annual Symposium on Foundations of Computer Science. IEEE Comput. Soc. Press., 124–134. https://doi.org/10.1109/sfcs.1994.365700 (1994)

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

    Article  MathSciNet  Google Scholar 

  44. Yadav, S., Islam, H., Prakash, O., Solé, P.: Self-dual and LCD double circulant and double negacirculant codes over \(\mathbb {F}_{q}+u\mathbb {F}_{q}+v\mathbb {F}_{q}.\) J. Appl. Math Comput. https://doi.org/10.1007/s12190-021-01499-9 (2021)

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

    Article  MathSciNet  Google Scholar 

  46. 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)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors are thankful to the University Grants Commission (UGC), Govt. of India for financial support under Sr. No. 2121540952, Ref. No. 20/12/2015(ii)EU-V dated 31/08/2016 and Indian Institute of Technology Patna for providing research facilities. We would also like to thank Prof. Patrick Solé (University Aix-Marseille, Marseille, France) and Prof. Smriti Singh (IIT Patna, India) for their careful reading and suggestions. Also, the authors would like to thank the anonymous referee(s) and the Editor for their valuable comments to improve the presentation of the paper.

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  1. Department of Mathematics, Indian Institute of Technology Patna, Patna, 801 106, India

    Habibul Islam & Om Prakash

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Islam, H., Prakash, O. Construction of LCD and new quantum codes from cyclic codes over a finite non-chain ring. Cryptogr. Commun. 14, 59–73 (2022). https://doi.org/10.1007/s12095-021-00516-9

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