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Quantum and LCD codes from skew constacyclic codes over a finite non-chain ring

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Abstract

For a prime p and a positive integer e, let \({\mathbb {F}}_{p^e}\) be the finite field and \( \mathfrak {R}:={\mathbb {F}}_{p^e}[u,v]/\langle f(u), g(v), uv - vu \rangle \), where f(u) and g(v) are non-constant square-free polynomials of degree r and s, respectively. This paper constructs quantum and linear complementary dual (briefly, LCD ) codes from skew constacyclic codes over the ring \(\mathfrak {R}\). Toward this, we first discuss the explicit structure of skew constacyclic codes and their Euclidean as well as Hermitian duals over \(\mathfrak {R}\). Then, we establish a necessary and sufficient condition for these codes to contain their Euclidean (Hermitian) duals. Further, by applying CSS (Hermitian) construction, many new quantum codes with better parameters are obtained. Moreover, a necessary and sufficient condition is established for these codes over \(\mathfrak {R}\) to be Euclidean (Hermitian) LCD. Finally, many examples of MDS codes over \({\mathbb {F}}_{p^e}\) are provided under the gray images of the skew Euclidean LCD codes.

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References

  1. Aly, S.A., Klappenecker, A., Sarvepalli, P.K.: On quantum and classical BCH codes. IEEE Trans. Inform. Theory 53(3), 1183–1188 (2007)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  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)

    MathSciNet  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(10), 4089–4098 (2016)

    ADS  MathSciNet  Google Scholar 

  5. Aydin, N., Guidotti, T. and Liu, P.: Good Classical and Quantum Codes from Multi-Twisted Codes (2020). arXiv preprint arXiv:2008.07037

  6. Boulanouar, R., Batoul, A., Boucher, D.: An overview on skew constacyclic codes and their subclass of LCD codes. Adv. Math. Commun. 15(4), 611–632 (2021)

    MathSciNet  MATH  Google Scholar 

  7. Berlekamp, E.R.: Algebraic Coding Theory. McGraw-Hill, New York (1968)

    MATH  Google Scholar 

  8. Bhardwaj, S., Raka, M.: Skew constacyclic codes over a non-chain ring \({\mathbb{F} }_q[u, v]/\langle f(u), g(v), uv-vu\rangle \). Appl. Algebra Engrg. Comm. Comput. 31(3), 173–194 (2020)

    MathSciNet  MATH  Google Scholar 

  9. Bosma, W. and Cannon, J.: Handbook of Magma Functions. Univ. of Sydney (1995)

  10. Boucher, D., Geiselmann, W., Ulmer, F.: Skew cyclic codes. Appl. Algebra Engrg. Comm. Comput. 18(4), 379–389 (2007)

    MathSciNet  MATH  Google Scholar 

  11. Boucher, D., Solé, P., Ulmer, F.: Skew constacyclic codes over Galois rings. Adv. Math. Commun. 2(3), 273–292 (2008)

    MathSciNet  MATH  Google Scholar 

  12. Boucher, D., Ulmer, F.: Coding with skew polynomial rings. J. Symb. Comput. 44(12), 1644–1656 (2009)

    MathSciNet  MATH  Google Scholar 

  13. Boucher, D. and Ulmer, F.: A note on the dual codes of module skew codes. In: IMA International Conference on Cryptography and Coding, pp. 230–243. Springer, Berlin (2011)

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  17. Dertli, A., Cengellenmis, Y., Eren, S.: On quantum codes obtained from cyclic codes over \(A_2\). Int. J. Quantum Inf. 13(3), 1–9 (2015)

    MATH  Google Scholar 

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

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

    MathSciNet  MATH  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 Eng. Commun. Comput. 30(2), 161–174 (2019)

    Google Scholar 

  21. Gao, J., Ma, F., Fu, F.: Skew constacyclic codes over the ring \({\mathbb{F} }_q+v{\mathbb{F} }_q\). J. Appl. Comput. Math. 6(3), 286–295 (2017)

    Google Scholar 

  22. 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(1), 1–9 (2018)

    ADS  MathSciNet  Google Scholar 

  23. Gao, J., Wang, Y.: Quantum codes derived from negacyclic codes. Int. J. Theor. Phys. 57(3), 682–686 (2018)

    MathSciNet  MATH  Google Scholar 

  24. Grassl, M., Beth, T., Rötteler, M.: On optimal quantum codes. Int. J. Quantum Inf. 2(01), 55–64 (2004)

    MATH  Google Scholar 

  25. Grassl, M. and Rötteler, M.: Quantum MDS codes over small fields. In: IEEE International Symposium on Information Theory (ISIT), pp. 1104-1108 (2015)

  26. La Guardia, G.G.: Quantum codes derived from cyclic codes. Internat. J. Theoret. Phys. 56(8), 2479–2484 (2017)

    ADS  MATH  Google Scholar 

  27. Gursoy, F., Siap, I., Yildiz, B.: Construction of skew cyclic codes over \({\mathbb{F} }_q+v{\mathbb{F} }_q\). Adv. Math. Commun. 8(3), 313–322 (2014)

    MathSciNet  Google Scholar 

  28. Hu, L., Yue, Q., He, X.: Quantum MDS codes from BCH constacyclic codes. Quantum Inf. Process. 17(12), 1–16 (2018)

    MathSciNet  MATH  Google Scholar 

  29. 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), 625–635 (2019)

    MathSciNet  MATH  Google Scholar 

  30. Islam, H., Verma, R.K., Prakash, O.: A family of constacyclic codes over \({\mathbb{F} }_{p^m}[u, v]/\langle u^{2}-1, v^{2}-1, uv-vu\rangle \). Int. J. Inf. Coding Theory 5(2), 101–116 (2019)

    Google Scholar 

  31. Rani, S., Verma, R.K., Prakash, O.: Quantum Codes from Repeated-Root Cyclic and Negacyclic Codes of Length \(4p^s\) Over \({\mathbb{F} }_{p^{m}}\). Internat. J. Theoret. Phys. 60(4), 1299–1327 (2021)

    ADS  MathSciNet  MATH  Google Scholar 

  32. Islam, H., Prakash, O., Verma, R.K.: New quantum codes from constacyclic codes over the ring \(R_{k, m}\). Adv. Math. Commun. 16(1), 17–35 (2022)

    MathSciNet  MATH  Google Scholar 

  33. Hammons, A.R., 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  MATH  Google Scholar 

  34. Jitman, S., Ling, S., Udomkavanich, P.: Skew constacyclic codes over finite chain ring. Adv. Math. Commun. 6(1), 39–63 (2012)

    MathSciNet  MATH  Google Scholar 

  35. Ore, O.: Theory of non-commutative polynomials. Ann. Math. 1, 480–508 (1933)

    MathSciNet  MATH  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(02), 689–700 (2011)

    MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  38. Li, C.: Hermitian LCD codes from cyclic codes. Des. Codes Cryptogr. 86(10), 2261–2278 (2018)

    ADS  MathSciNet  MATH  Google Scholar 

  39. 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 (2022)

    Google Scholar 

  40. Li, J., Gao, J., Wang, Y.: Quantum codes from \((1--2v)\)-constacyclic codes over the ring \({\mathbb{F} }_q+u{\mathbb{F} }_q+v{\mathbb{F} }_q+uv{\mathbb{F} }_q\). Discrete Math. Algorithms Appl. 10(4), 1–8 (2018)

  41. Liu, H., Liu, X.: Constructions of quantum MDS codes. Quantum Inf. Process. 20(14), 1–13 (2021)

    ADS  MathSciNet  MATH  Google Scholar 

  42. Liu, Z., Wang, J.: Linear complementary dual codes over rings. Des. Codes Cryptogr. 87(12), 3077–3086 (2019)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  44. Ma, F., Gao, J., Fu, F.W.: Constacyclic codes over the ring \({\mathbb{F} }_{p}+v{\mathbb{F} }_{p}+v^{2}{\mathbb{F} }_{p}\) and their applications of constructing new non-binary quantum codes. Quantum Inf. Process. 17(6), 1–19 (2018)

    MathSciNet  Google Scholar 

  45. Ma, F., Gao, J., Fu, F.W.: 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(2), 421–434 (2019)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  47. Ozen, M., Ozzaim, T., Ince, H.: Skew quasi cyclic codes over \({\mathbb{F} }_q+v{\mathbb{F} }_q\). J. Algebra Appl. 18(4), 1–16 (2019)

    Google Scholar 

  48. Prakash, O., Yadav, S., Verma, R.K.: Constacyclic and Linear Complementary Dual Codes Over \({\mathbb{F} }_q+u{\mathbb{F} }_q\). Defence Sci. J. 70(6), 626–632 (2020)

    Google Scholar 

  49. Prange, E.: Cyclic error-correcting codes in two symbols. Air force Cambridge research center (1957)

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

    MATH  Google Scholar 

  51. Rains, E.M.: Nonbinary quantum codes. IEEE Trans. Inform. Theory 45(06), 1827–1832 (1999)

    MathSciNet  MATH  Google Scholar 

  52. Rains, E.M.: Quantum codes of minimum distance two. IEEE Trans. Inform. Theory 45(01), 266–271 (1999)

    MathSciNet  MATH  Google Scholar 

  53. Sharma, A., Chauhan, V.: Skew multi-twisted codes over finite fields and their Galois duals. Finite Fields Appl. 59, 297–334 (2019)

    MathSciNet  MATH  Google Scholar 

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

    ADS  Google Scholar 

  55. Verma, R.K., Prakash, O., Islam, H., Singh, A.: New non-binary quantum codes from skew constacyclic and additive skew constacyclic codes. Eur. Phys. J. Plus 137(2), 1–13 (2022)

    Google Scholar 

  56. Verma, R.K., Prakash, O. and Singh, A.: Quantum codes from skew constacyclic codes over \(F_{p^m}+ vF_{p^m}+ v^2 F_{p^m}\). 2020 Algebraic and Combinatorial Coding Theory (ACCT), pp. 156-161. IEEE (2020)

  57. Verma, R.K., Prakash, O., Singh, A., Islam, H.: New quantum codes from skew constacyclic codes. Adv. Math. Commun. 17(4), 900–919 (2023)

    MathSciNet  Google Scholar 

  58. 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), 1–17 (2021)

    ADS  Google Scholar 

  59. 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  MATH  Google Scholar 

  60. Yao, T., Shi, M., Solé, P.: Skew cyclic codes over \({\mathbb{F} }_q+{\mathbb{F} }_q+v{\mathbb{F} }_q+uv{\mathbb{F} }_q\). J. Algebra Comb. Discrete Struct. Appl. 2(3), 163–168 (2015)

    MathSciNet  Google Scholar 

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Acknowledgements

The authors are thankful to the DST, Govt. of India (under CRG/2020/005927, vide Diary No. SERB/F/6780/ 2020-2021 dated 31 December, 2020) and the CSIR, Govt. of India (under grant no. 09/1023(0014)/2015-EMR-I and 09/1023(0027)/2019-EMR-I) for providing financial support. The authors would also like to thank the anonymous referee(s) and the Editor for their valuable comments to improve the presentation of the manuscript.

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

    Om Prakash, Ram Krishna Verma & Ashutosh Singh

  2. SRM Institute of Science and Technology, Delhi-NCR Campus, Ghaziabad, Uttar Pradesh, 201204, India

    Ram Krishna Verma

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  1. Om Prakash
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Prakash, O., Verma, R.K. & Singh, A. Quantum and LCD codes from skew constacyclic codes over a finite non-chain ring. Quantum Inf Process 22, 200 (2023). https://doi.org/10.1007/s11128-023-03951-0

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