Skip to main content
SpringerLink
Log in

Construction of one-gray weight codes and two-Gray weight codes over ℤ4 + uℤ4

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

Abstract

This paper firstly gives some necessary conditions on one-Gray weight linear codes. And then we use these results to construct several classes of one-Gray weight linear codes over ℤ4+uℤ4(u 2 = u) with type \({16^{{k_1}}}{8^{{k_2}}}{8^{{k_3}}}{4^{{k_4}}}{4^{{k_5}}}{4^{{k_6}}}{2^{{k_7}}}{2^{{k_8}}}\) based on a distance-preserving Gray map from (ℤ4 + uℤ4)n to ℤ 2n4 . Secondly, the authors use the similar approach to do works on two-Gray (projective) weight linear codes. Finally, some examples are given to illustrate the construction methods.

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: 18–181.

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. MacWilliams F J and Sloane J A, The theory of error-correcting codes, Proceedings of the IEEE, 1980, 68(1): 185–186.

    MATH  Google Scholar 

  4. 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  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

  9. Carlet C, One-Weight Z 4-Linear Codes, Coding Theory, Cryptography and Related Areas, Springer, Berlin Heidelberg, 2000, 57–72.

    Book  Google Scholar 

  10. 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 

  11. Shi M J, Zhu S X, and Yang S L, A class of optimal p-ary codes from one-weight codes over F p[u]/<u m>, J. Frank. Inst., 2013, 350(5): 929–937.

    Article  MathSciNet  MATH  Google Scholar 

  12. 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.

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Yuan J and Ding C S, Secret sharing schemes from two-weight codes, Electronic Notes in Discrete Mathematics, 2003, 15: 232.

    Article  MathSciNet  MATH  Google Scholar 

  17. Shi M J and Chen L, Construction of two-Lee weight codes over \(\mathbb{F}_p + v\mathbb{F}_p + v^2 \mathbb{F}_p\), International Journal of Computer Mathematics, 2016, 93(3): 415–424.

    Article  MathSciNet  MATH  Google Scholar 

  18. Shi M J and Solé P, Optimal p-ary codes from one-weight and two-weight codes over \(\mathbb{F}_p + v\mathbb{F}_p\), Jornal of System Science and Complexity, 2015, 28(3): 679–690.

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Gao J and Gao Y, Some results on linear codes over ℤ4 + vℤ4, arXiv: 1402.6771, 2014.

    Google Scholar 

  21. Bandi R K and Bhaintwal M. Codes over ℤ4+vℤ4 with respect to Rosenbloom-Tsfasman metirc, IEEE, ICACCI, 2013, 37–41.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Key Laboratory of Intelligent Computing & Signal Processing, Ministry of Education, Anhui University, Hefei, 230039, China

    Minjia Shi & Dandan Wang

  2. National Mobile Communications Research Laboratory, Southeast University, Nanjing, 210096, China

    Minjia Shi & Dandan Wang

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

    Minjia Shi, Dandan Wang & Bo Wu

  4. School of Science, Shandong University of Technology, Zibo, 255091, China

    Jian Gao

  5. State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing, 100093, China

    Bo Wu

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. Jian Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Bo Wu
    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 Nos. 61672036 and 61202068, the Open Research Fund of National Mobile Communications Research Laboratory, Southeast University under Grant No. 2015D11, Technology Foundation for Selected Overseas Chinese Scholar, Ministry of Personnel of China under Grant No. 05015133 and Key projects of support program for outstanding young talents in Colleges and Universities under Grant No. gxyqZD2016008.

This paper was recommended for publication by Editor ZHANG Zhifang.

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., Gao, J. et al. Construction of one-gray weight codes and two-Gray weight codes over ℤ4 + uℤ4 . J Syst Sci Complex 29, 1472–1484 (2016). https://doi.org/10.1007/s11424-016-5286-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-016-5286-y

Keywords

Search

Navigation