Skip to main content
Log in

MacDonald codes over the ring \({\mathbb {F}}_{p}+v{\mathbb {F}}_{p}+v^2{\mathbb {F}}_{p}\)

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we consider MacDonald codes over the finite non-chain ring \({\mathbb {F}}_p+v{\mathbb {F}}_p+v^2{\mathbb {F}}_p\) and their applications in constructing secret sharing schemes and association schemes, where p is an odd prime and \(v^3=v\). We give some structural properties of MacDonald codes first. Then, we study the weight enumerators of torsion codes of these MacDonald codes. As some applications, constructing secret sharing schemes and association schemes is also investigated.

This is a preview of subscription content, log in via an institution to check access.

Access this article

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

  • Ashikhmin A, Barg A (1998) Minimal vectors in linear codes. IEEE Trans Inf Theory 44(5):2010–2017

    Article  MathSciNet  Google Scholar 

  • Bosma W, Cannon J, Playoust C (1997) The Magma algebra system I: the user language. J Symb Comput 24(3):235–265

    Article  MathSciNet  Google Scholar 

  • Colbourn, C, Gupta, M (2003) On quaternary MacDonald codes. In: Proceeding of IEEE international conference on information technology: coding and computing. Las Vegas, pp 212–215

  • Delsarte P (1973) An algebraic approach to the association schemes of coding theory. J Philips Res Rep Suppl 10:97

    MathSciNet  MATH  Google Scholar 

  • Dertli A, Cengellenmis Y (2011) MacDonald codes over the ring \({\mathbb{F}}_2+v{\mathbb{F}}_2\). Int J Algebra 5(20):985–991

    Google Scholar 

  • Ding C, Yuan J (2003) Covering and secret sharing with linear codes. Discrete mathematics and theoretical computer science. Springer, LNCS 2731, Berlin, Heidelberg, pp 11–25

  • Gao J (2015) Some results on linear codes over \({\mathbb{F}}_p+u{\mathbb{F}}_p+u^2{\mathbb{F}}_p\). J Appl Math Comput 47(1–2):473–485

    Google Scholar 

  • Luo G, Cao X, Xu G, Xu S (2018) A new class of optimal linear codes with flexible parameters. Discrete Appl Math 237:126–131

    Article  MathSciNet  Google Scholar 

  • MacDonald J (1960) Design methods for maximum minimum distance errorcorrecting codes. IBM J Res Dev 4:43–57

    Article  Google Scholar 

  • Massey JL (1993) Minimal codewords and secret sharing. In: Proceedings of the 6th Joint Swedish-Russian workshop on information theory, Netherlands, Veldhoven, pp 276–279

  • Patel A (1975) Maximal \(q\)-ary linear codes with large minimum distance. IEEE Trans Inf Theory 21:106–110

    Google Scholar 

  • Shi M, Guan Y, Solé P (2017a) Two new families of two-weight codes. IEEE Trans Inf Theory 63(10):6240–6246

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

    Google Scholar 

  • Shi M, Xu L, Yang G (2017b) A note on one weight and two weight projective \({\mathbb{Z}}_4\)-codes. IEEE Trans Inf Theory 63(1):177–182

  • Wang X, Gao J, Fu F-W (2016) Secret sharing schemes from linear codes over \({\mathbb{F}}_p+v{\mathbb{F}}_p\). Int J Found Comput Sci 27(5):595–605

    Google Scholar 

Download references

Acknowledgements

This research is supported by the National Natural Science Foundation of China (Grant Nos. 11701336, 11626144 and 11671235), and the Scientific Research Fund of Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering (No. 2018MMAEZD04).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jian Gao.

Additional information

Communicated by Thomas Aaron Gulliver.

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, Y., Gao, J. MacDonald codes over the ring \({\mathbb {F}}_{p}+v{\mathbb {F}}_{p}+v^2{\mathbb {F}}_{p}\). Comp. Appl. Math. 38, 169 (2019). https://doi.org/10.1007/s40314-019-0937-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-019-0937-y

Keywords

Mathematics Subject Classification

Navigation