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

  • Yongkang Wang
  • Jian GaoEmail author


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.


MacDonald codes Torsion codes Secret sharing schemes Association schemes 

Mathematics Subject Classification

94B05 11T71 



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).


  1. Ashikhmin A, Barg A (1998) Minimal vectors in linear codes. IEEE Trans Inf Theory 44(5):2010–2017MathSciNetCrossRefGoogle Scholar
  2. Bosma W, Cannon J, Playoust C (1997) The Magma algebra system I: the user language. J Symb Comput 24(3):235–265MathSciNetCrossRefGoogle Scholar
  3. 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 Google Scholar
  4. Delsarte P (1973) An algebraic approach to the association schemes of coding theory. J Philips Res Rep Suppl 10:97MathSciNetzbMATHGoogle Scholar
  5. Dertli A, Cengellenmis Y (2011) MacDonald codes over the ring \({\mathbb{F}}_2+v{\mathbb{F}}_2\). Int J Algebra 5(20):985–991Google Scholar
  6. Ding C, Yuan J (2003) Covering and secret sharing with linear codes. Discrete mathematics and theoretical computer science. Springer, LNCS 2731, Berlin, Heidelberg, pp 11–25Google Scholar
  7. 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):473–485Google Scholar
  8. Luo G, Cao X, Xu G, Xu S (2018) A new class of optimal linear codes with flexible parameters. Discrete Appl Math 237:126–131MathSciNetCrossRefGoogle Scholar
  9. MacDonald J (1960) Design methods for maximum minimum distance errorcorrecting codes. IBM J Res Dev 4:43–57CrossRefGoogle Scholar
  10. Massey JL (1993) Minimal codewords and secret sharing. In: Proceedings of the 6th Joint Swedish-Russian workshop on information theory, Netherlands, Veldhoven, pp 276–279Google Scholar
  11. Patel A (1975) Maximal \(q\)-ary linear codes with large minimum distance. IEEE Trans Inf Theory 21:106–110Google Scholar
  12. Shi M, Guan Y, Solé P (2017a) Two new families of two-weight codes. IEEE Trans Inf Theory 63(10):6240–6246Google Scholar
  13. 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):247–255Google Scholar
  14. 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):177–182Google Scholar
  15. 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):595–605Google Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsShandong University of TechnologyZiboPeople’s Republic of China
  2. 2.School of Mathematics and StatisticsChangsha University of Science and TechnologyChangshaPeople’s Republic of China

Personalised recommendations