Skip to main content
SpringerLink
Log in

Optimal p-ary codes from one-weight and two-weight codes over \(\mathbb{F}_p + v\mathbb{F}_p^* \)

  • Published:
Journal of Systems Science and Complexity Aims and scope Submit manuscript

Abstract

This paper is devoted to determining the structures and properties of one-Lee weight codes and two-Lee weight projective codes \(C_{k_1 ,k_2 ,k_3 } \) over \(\mathbb{F}_p + v\mathbb{F}_p^* \) with type \(p^{2k_1 } p^{k_2 } p^{k_3 } \). The authors introduce a distance-preserving Gray map from \((\mathbb{F}_p + v\mathbb{F}_p )^n \) to \(\mathbb{F}_p^{2n} \). By the Gray map, the authors construct a family of optimal one-Hamming weight p-ary linear codes from one-Lee weight codes over \(\mathbb{F}_p + v\mathbb{F}_p \), which attain the Plotkin bound and the Griesmer bound. The authors also obtain a class of optimal p-ary linear codes from two-Lee weight projective codes over \(\mathbb{F}_p + v\mathbb{F}_p \), which meet the Griesmer bound.

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

    MATH  MathSciNet  Google Scholar 

  2. Lint J V and Tolhuizenon L, On perfect ternary constant weight codes, Designs, Codes and Cryptography, 1999, 18(1–3): 231–234.

    Article  MATH  Google Scholar 

  3. MacWilliams F J and N Sloane J A, The Theory of Error-Correcting Codes, Amsterdam, Nethland, 1977.

    MATH  Google Scholar 

  4. Carlet C, One-weight ℤ4-Linear Codes, Coding Theory, Cryptography and Related Areas, Springer, 2000, 57–72.

    Book  Google Scholar 

  5. Wood J A, The structure of linear codes of constant weight, Trans. Amer. Math. Soc., 2001, 354(3): 1007–1026.

    Article  Google Scholar 

  6. Shi M J, Optimal p-ary codes from one-weight linear codes over \(\mathbb{Z}_{p^m } \), Chinese Journal of Electronics, 2013, 22(4): 799–802.

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  10. Brouwer A E and Eupen M V, The correspondence between projective codes and 2-weight codes, Designs, Codes and Cryptography, 1997, 11(3): 261–266.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  12. Rao A and Pinnawala N, A family of two-weight irreducible cyclic codes, IEEE Trans. Inform. Theory, 2010, 56(6): 2568–257.

    Article  MathSciNet  Google Scholar 

  13. Schmidt B and White C, All two-weight irreducible cyclic codes? Finite Fields Appl., 2002, 8(1): 1–17.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  15. Vega G and Wolfmann J, New class of 2-weight cyclic codes, Designs, Codes and Cryptography, 2007, 42(3): 327–334.

    Article  MATH  MathSciNet  Google Scholar 

  16. Vega G, Determining the number of one-weight cyclic codes when length and dimension are given, Lecture Notes in Computer Science, 2007, 4547: 284–293.

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  18. Wolfmann J, Are 2-weight projective cyclic codes irreducible? IEEE Trans. Inform Theory, 2005, 51(2): 733–737.

    Article  MATH  MathSciNet  Google Scholar 

  19. Byrne E, M Greferath, and Honold T, Ring geometries, two-weight codes, and strongly regular graphs, Designs, Codes and Cryptography, 2008, 48(1): 1–16.

    Article  MATH  MathSciNet  Google Scholar 

  20. Ling S and Solé P, Two-weight codes over chain rings and partial difference sets, Rapport de recherche I3S/RR-2002-40FR, http://www.i3s.unice.fr/I3S/FR.

  21. Yuan J and Ding C S, Secret sharing schemes from two-weight codes, Bose Centenary Symposium on Discrete Mathematics and Applications, Indian Statistical Institute, Kolkata, India, 2002, 20–23.

    Google Scholar 

  22. Shi M J and Wang Y, Optimal binary codes from one-weight codes and two-weight projective codes over ℤ4, Journal of Systems Science and Complexity, 2014, 27(4): 795–810.

    Article  MathSciNet  Google Scholar 

  23. Shi M J, Yang S L, and Zhu S X, Good p-ary quasi-cyclic codes from cyclic codes over \(\mathbb{F}_p + v\mathbb{F}_p \), Journal of Systems Science and Complexity, 2012, 25(2): 375–384.

    Article  MATH  MathSciNet  Google Scholar 

  24. Ling S and Xing C P, Coding Theory — A First Course, Cambridge, Cambridge University Press, 2004, 83–95.

    Book  Google Scholar 

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

    Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Minjia Shi

  2. 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. Patrick Solé
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Minjia Shi.

Additional information

This research was supported by the National Natural Science Foundation of China under Grant No. 61202068, Talented youth Fund of Anhui Province Universities under Grant No. 2012SQRL020ZD, and the Technology Foundation for Selected Overseas Chinese Scholar, Ministry of Personnel of China under Grant No. 05015133.

This paper was recommended for publication by Editor HU Lei.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shi, M., 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, 679–690 (2015). https://doi.org/10.1007/s11424-015-3265-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-015-3265-3

Keywords

Search

Navigation