Skip to main content
SpringerLink
Log in

Linear codes over \({\mathbbm {F}}_3+u\mathbbm {F}_3+u^2\mathbbm {F}_3\): MacWilliams identities, optimal ternary codes from one-Lee weight codes and two-Lee weight codes

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

Abstract

Let \(R=\mathbbm {F}_3+u\mathbbm {F}_3+u^2\mathbbm {F}_3\), where \(u^3=1\). Lee weights, Gray map for linear codes over R are defined in this paper, and the MacWilliams identities for complete, Hamming, symmetrized and Lee weight enumerators are verified. Moreover, the construction method of one-Lee weight codes over R with type \(27^{k_1}9^{k_2}3^{k_3}\) is determined. By the Gray map, a family of one-weight ternary linear codes is obtained, whose parameters attain the Plotkin bound and Griesmer bound. We also obtain a class of optimal ternary codes from two-Lee weight projective codes over R, which meet the Griesmer bound. Finally, some examples are given to illustrate the results.

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. Bahattin, Y., Karadeniz, S.: Linear code over \({\mathbb{Z}}_4+u{\mathbb{Z}}_4\): MacWilliams identitise, projections, and formally self-dual codes. Finite Field Appl. 27, 24–40 (2014)

    Article  MATH  Google Scholar 

  2. Bonisoli, A.: Every equidistant linear code is a sequence of dual Hamming codes. Ars Comb. 18, 181–186 (1984)

    MathSciNet  MATH  Google Scholar 

  3. Bouyukliev, I., Fack, V., Willems, W., Winne, J.: Projective two-weight codes with small parameters and their corresponding graphs. Des. Codes Cryptogr. 41(1), 59–78 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Brouwer, A.E., Eupen, M.V.: The correspondence between projective codes and \(2\)-weight codes. Des. Codes Cryptogr. 11(3), 261–266 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. Calderbank, R., Kantor, W.M.: The geometry of two-weight codes. Bull. Lond. Math. Soc. 18(2), 97–122 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  6. Carlet, C.: One-Weight \({\mathbb{Z}}_{4}\)-Linear Codes. CodingTheory, Cryptography and Related Areas. Springer, New York (2000)

    Google Scholar 

  7. Gao, J., Gao, Y., Fu, F.W.: Some Results on Linear Codes Over \({\mathbb{Z}}_4+v{\mathbb{Z}}_4\). arXiv:1402.6771 (2014)

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Kaya, A., Yildiz, B.: New extremal binary selef-dual code of length 68. J. Algebra Comb. Discret. Struct. Appl. 1(1), 29–39 (2014)

    Google Scholar 

  10. Lint, J.V., Tolhuizenon, L.: On perfect ternary constant weight codes. Des. Codes Cryptogr. 18(1–3), 231–234 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. Holland Publishing Company, Amsterdam (1977)

    MATH  Google Scholar 

  12. Shi M. J., Chen L.: Construction of two-Lee weight codes over \({\mathbb{F}}_p+v{\mathbb{F}}_p+v^{2}{\mathbb{F}}_p\), Int. J. Comput. Math. (2014). doi:10.1080/00207160.2014.989709

  13. Shi, M.J., Solé, P.: Optimal \(p\)-ary codes from one-weight and two-weight codes over \(\mathbb{F} _p+ v\mathbb{F} _p\). J. Syst. Sci. Complex. 28(3), 679–690 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Shi, M.J., Solé, P., Wu, B.: Cyclic codes and the weight enumerator of linear codes over \({\mathbb{F}}_2+v{\mathbb{F}}_2+v^2{\mathbb{F}}_2\). Appl. Comput. Math. 12(2), 247–255 (2013)

    MathSciNet  Google Scholar 

  15. Shi, M.J., Yang, S.L.: MacWilliams identities of linear codes over non-principal ring \({\mathbb{F}}_p+v{\mathbb{F}}_p\). Acta Eletron. Sin. 39(10), 2449–2453 (2011)

    Google Scholar 

  16. Shi, M.J., Zhu, S.X., Yang, S.L.: A class of optimal \(p\)-ary codes from one-weight codes over \({\mathbb{F}}_{p}[u]/\langle u^m\rangle \). J. Frankl. Inst. 350(5), 929–937 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Shi, M.J., Wang, Y.: Optimal binary codes from one-Lee weight and two-Lee weight projective codes over \({\mathbb{Z}}_4\). J. Syst. Sci. Complex. 27(4), 795–810 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Van Lint, J.H., Schrijver, A.: Construction of strongly regular graphs, two-weight codes and partial geometries by finite fields. Combinatorica 1(1), 63–73 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  19. Vega, G.: Determining the number of one-weight cyclic codes when length and dimension are given. LNCS 4547, 284–293 (2007)

    MathSciNet  MATH  Google Scholar 

  20. Vega, G.: Two-weight cyclic codes constructed as the direct sum of two one-weight cyclic codes. Fnite Fields Appl. 14(3), 785–797 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Wan, Z.X.: Quaternary Codes. World Scientific, Singapore (1997)

    MATH  Google Scholar 

  22. Wolfmann, J.: Projective two-weight irreducible cyclic and constacyclic codes. Fnite Fields Appl. 14(2), 351–360 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Wood, J.A.: The structure of linear codes of constant weight. Trans. Am. Math. Soc. 2001(354), 1007–1026 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the anonymous referees for their valuable comments and suggestions, which greatly improved the final presentation of the manuscript. This work is supported by National Natural Science Foundation of China (61202068) and Technology Foundation for Selected Overseas Chinese Scholar, Ministry of Personnel of China (05015133).

Author information

Authors and Affiliations

  1. Key Laboratory of Intelligent Computing & Signal Processing Ministry of Education, Anhui University, No. 3 Feixi Road, Hefei, 230039, Anhui Province, People’s Republic of China

    Minjia Shi

  2. School of Mathematical Sciences, Anhui University, Hefei, 230601, Anhui, China

    Minjia Shi & Dandan Wang

  3. CNRS/LTCI Telecom ParisTech, Paris, France

    Patrick Solé

Authors
  1. Minjia Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dandan Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Patrick Solé
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Minjia Shi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shi, M., Wang, D. & Solé, P. Linear codes over \({\mathbbm {F}}_3+u\mathbbm {F}_3+u^2\mathbbm {F}_3\): MacWilliams identities, optimal ternary codes from one-Lee weight codes and two-Lee weight codes. J. Appl. Math. Comput. 51, 527–544 (2016). https://doi.org/10.1007/s12190-015-0918-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-015-0918-2

Keywords

Search

Navigation