Skip to main content
SpringerLink
Log in

On \(\mathbb {Z}_2\mathbb {Z}_4\)-additive polycyclic codes and their Gray images

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

In this paper, we first generalize the polycyclic codes over finite fields to polycyclic codes over the mixed alphabet \(\mathbb {Z}_2\mathbb {Z}_4\), and we show that these codes can be identified as \(\mathbb {Z}_4[x]\)-submodules of \(\mathcal {R}_{\alpha ,\beta }\) with \(\mathcal {R}_{\alpha ,\beta }=\mathbb {Z}_2[x]/\langle t_1(x)\rangle \times \mathbb {Z}_4[x]/\langle t_2(x)\rangle \), where \(t_1(x)\) and \(t_2(x)\) are monic polynomials over \(\mathbb {Z}_2\) and \(\mathbb {Z}_4\), respectively. Then we provide the generator polynomials and minimal generating sets for this family of codes based on the strong Gröbner basis. In particular, under the proper defined inner product, we study the dual of \(\mathbb {Z}_2\mathbb {Z}_4\)-additive polycyclic codes. Finally, we focus on the characterization of the \(\mathbb {Z}_2\mathbb {Z}_4\)-MDSS and MDSR codes, and as examples, we also present some (almost) optimal binary codes derived from the \(\mathbb {Z}_2\mathbb {Z}_4\)-additive polycyclic codes.

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.

Similar content being viewed by others

References

  1. Abualrub T., Siap I., Aydin N.: \(\mathbb{Z}_2\mathbb{Z}_4\)-additive cyclic codes. IEEE Trans. Inf. Theory 60(3), 1508–1514 (2014).

    Article  MATH  Google Scholar 

  2. Alahmadi A., Dougherty S. T., Leroy A., Solé P.: On the duality and the direction of polycyclic codes. Adv. Math. Commun. 10(4), 921–929 (2016).

  3. Aydogdu I., Siap I.: The structure of \(\mathbb{Z}_2\mathbb{Z}_{2^s}\)-additive codes: bounds on the minimum distance. Appl. Math. Inf. Sci. 7(6), 2271–2278 (2013).

    Article  MathSciNet  Google Scholar 

  4. Aydogdu I., Siap I.: On \(\mathbb{Z}_{p^r}\mathbb{Z}_{p^s}\)-additive codes. Linear Multilinear Algeb. 63(10), 2089–2102 (2015).

    Article  MATH  Google Scholar 

  5. Aydogdu I., Abualrub T., Siap I.: \(\mathbb{Z}_2\mathbb{Z}_2[u]\)-cyclic and constacyclic codes. IEEE Trans. Inf. Theory 63(8), 4883–4893 (2017).

    Article  MATH  Google Scholar 

  6. Benbelkacema N., Borges J., Dougherty S.T., Fernández-Córdoba C.: On \(\mathbb{Z}_2\mathbb{Z}_4\)-additive complementary dual codes and related LCD codes. Finite Fields Appl. 62, 101622 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  7. Berger T.P., Amrani N.E.: Codes over finite quotients of polynomial ring. Finite Fields Appl. 25, 165–181 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  8. Bilal M., Borges J., Dougherty S.T., Fernández-Cárdoba C.: Maximum distance separable codes over \(\mathbb{Z}_4\) and \(\mathbb{Z}_2\times \mathbb{Z}_4\). Des. Codes Cryptogr. 61(1), 31–40 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  9. Borges J., Fernández-Córdoba C.: A characterization of \(\mathbb{Z}_2\mathbb{Z}_2[u]\)-linear codes. Des. Codes Cryptogr. 86(7), 1377–1389 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  10. Borges J., Rifà J.: A characterization of 1-perfect additive codes. IEEE Trans. Inf. Theory 45(5), 1688–1697 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  11. Borges J., Fernández-Córdoba C., Pujol J., Rifà J., Villanueva M.: \(\mathbb{Z}_2\mathbb{Z}_4\)-linear codes: generator matrices and duality. Des. Codes Cryptogr. 54(2), 167–179 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  12. Borges J., Fernández-Córdoba C., Ten-Valls R.: \(\mathbb{Z}_2\mathbb{Z}_4\)-additive cyclic codes, generator polynomials, and dual codes. IEEE Trans. Inf. Theory 62(11), 6348–6354 (2016).

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Delsarte P.: An Algebraic Approach to the Association Schemes of Coding Theory, (Philips Research Reports). Historical Jrl, Ann Arbor (1973).

    MATH  Google Scholar 

  15. Dougherty S.T., Liu H.W., Yu L.: One weight \(\mathbb{Z}_2\mathbb{Z}_4\)-additive codes. AAECC 27(2), 123–138 (2016).

    Article  Google Scholar 

  16. Fan Y., Liu H.L.: \(\mathbb{Z}_2\mathbb{Z}_4\)-additive cyclic codes are asymptotically good, (2019). arXiv:1911.09350.

  17. Fotue-Tabue A., Martíez-Moro E., Blackford J.T.: On polycyclic codes over a finite chain ring. Adv. Math. Commun. 14(3), 455–466 (2020).

    MathSciNet  MATH  Google Scholar 

  18. Grassl M.: Table of bounds on linear codes. (1995). [Online]. http://www.codetables.de/.

  19. 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. Inf. Theory 40(2), 301–319 (1994).

    Article  MATH  Google Scholar 

  20. Hou X.T., Gao J.: \(\mathbb{Z}_p\mathbb{Z}_p[v]\)-additive cyclic codes are asymptotically good. J. Appl. Math. Comput. 66(1-2), 871–884 (2021).

  21. Hu P., Liu X.S.: Binary LCD codes from \(\mathbb{Z}_2\mathbb{Z}_2[u]\) (2019). arXiv:1903.11380v1.

  22. López-Permouth S.R., Parra-Avila B.R., Szabo S.: Dual generalizations of the concept of cyclicity of codes. Adv. Math. Commun. 3(3), 227–234 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  23. López-Permouth S.R., Özadam H., Özbudak F., Szabo S.: Polycyclic codes over Galois rings with applications to repeated-root constacyclic codes. Finite Fields Appl. 19(1), 16–38 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  24. Mahmoudi S., Samei K.: Additive codes over Galois rings. Finite Fields Appl. 56, 332–350 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  25. Martíez-Moro E., Rúa I.F.: Multivariable codes over finite chain rings: serial codes. SIAM J. Discrete Math. 20(4), 947–959 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  26. Martíez-Moro E., Rúa I.F.: On repeated-root multivariable codes over a finite chain rings. Des. Codes Cryptogr. 45(2), 219–227 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  27. Martíez-Moro E., Piñera-Nicolás A., Rúa I.F.: Codes over affine algebras with a finite communicative chain coefficient ring. Finite Fields Appl. 49, 94–107 (2018).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  29. Nikseresht A.: Dual of codes over finite quotients of polynomial rings. Finite Fields Appl. 45, 323–340 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  30. Norton G.H., Sǎlǎgean A.: Strong Gröbner basis for polynomials over a principal ideal ring. Bull. Austral. Math. Soc. 64(3), 505–528 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  31. Norton G.H., Sǎlǎgean A.: Cyclic codes and minimal strong Gröbner basis over a principal ideal ring. Finite Fields Appl. 9(2), 237–249 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  32. Peterson W.W., Weldon E.J.: Error-Correcting Codes, 2nd edn. MIT Press, Cambridge (1972).

    MATH  Google Scholar 

  33. Qian L.Q., Cao X.W.: Bounds and optimal \(q\)-ary codes derived from the \(\mathbb{Z}_qR\)-cyclic codes. IEEE Trans. Inf. Theory 66(2), 923–935 (2020).

    Article  MATH  Google Scholar 

  34. Rifà J., Pujol J.: Translation invariant propeliniear codes. IEEE Trans. Inf. Theory 43(2), 590–598 (1997).

    Article  MATH  Google Scholar 

  35. Sǎlǎgean A.: Repeated-root cyclic and nagacyclic codes over a finite chain ring. Discrete Appl. Math. 154(2), 413–419 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  36. Shi M.J., Wu R.S., Krotov D.S.: On \(\mathbb{Z}_p\mathbb{Z}_{p^k}\)-additive codes and their dality. IEEE Trans. Inf. Theory 65(6), 3841–3847 (2019).

    Article  MATH  Google Scholar 

  37. Shi M.J., Wang C.C., Wu R.S., Hu Y., Chang Y.Q.: One-weight and two-weight \(\mathbb{Z}_2\mathbb{Z}_2[u, v]\)-additive codes. Cryptogr. Commun. 12(3), 443–454 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  38. Wood J.A.: Duality for modules over finite rings and applications to coding theory. Am. J. Math. 121(3), 555–575 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  39. Wood J.A.: Lecture notes on the MacWilliams identities and extension theorem, Lectures for the Simat International School and Conference on Coding Theory, November 30–December 2, (2008).

  40. Yao T., Zhu S.X.: \(\mathbb{Z}_p\mathbb{Z}_{p^s}\)-additive cyclic codes are asymptotically good. Cryptogr. Commun. 12(2), 253–264 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  41. Yao T., Zhu S.X., Kai X.S.: Asymptotically good \(\mathbb{Z}_{p^r}\mathbb{Z}_{p^s}\)-additive cyclic codes. Finite Fields Appl. 63, 101633 (2020).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Key Laboratory of Intelligent Computing and Signal Processing, Ministry of Education, School of Mathematical Sciences, Anhui University, Hefei, 230601, Anhui, P.R. China

    Rongsheng Wu & Minjia Shi

Authors
  1. Rongsheng Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Minjia Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Minjia Shi.

Additional information

Publisher's Note

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

This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue: On Coding Theory and Combinatorics: In Memory of Vera Pless”

This research is supported by the National Natural Science Foundation of China (Grant Nos. 12071001, 61672036), the Excellent Youth Foundation of Natural Science Foundation of Anhui Province (Grant No. 1808085J20).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wu, R., Shi, M. On \(\mathbb {Z}_2\mathbb {Z}_4\)-additive polycyclic codes and their Gray images. Des. Codes Cryptogr. 90, 2551–2562 (2022). https://doi.org/10.1007/s10623-021-00917-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-021-00917-0

Keywords

Search

Navigation