Designs, Codes and Cryptography

, Volume 86, Issue 5, pp 1149–1159 | Cite as

Cyclotomic construction of strong external difference families in finite fields

  • Jiejing Wen
  • Minghui Yang
  • Fangwei Fu
  • Keqin Feng


Strong external difference families (SEDFs) and their generalizations GSEDFs and BGSEDFs in a finite abelian group G are combinatorial designs introduced by Paterson and Stinson (Discret Math 339: 2891–2906, 2016) and have applications in communication theory to construct optimal strong algebraic manipulation detection codes. In this paper we firstly present some general constructions of these combinatorial designs by using difference sets and partial difference sets in G. Then, as applications of the general constructions, we construct series of SEDF, GSEDF and BGSEDF in finite fields by using cyclotomic classes. Particularly, we present an \((n,m,k,\lambda )=(243,11,22,20)\)-SEDF in \((\mathbb {F}_q,+)\ (q=3^5=243)\) by using the cyclotomic classes of order 11 in \(\mathbb {F}_q\) which answers an open problem raised in Paterson and Stinson (2016).


Strong external difference family Difference set Partial difference set Cyclotomic class Cyclotomic number Finite field Strong algebraic manipulation detection code 

Mathematics Subject Classification

05B10 11T22 



The authors are grateful to the two anonoymous reviewers for their detailed comments and suggestions that much improved the presentation and quality of this paper. The work of J. Wen and F. Fu was supported by the National Key Basic Research Program of China under Grant 2013CB834204, and the NSFC under Grant 61571243, 61171082. The work of M. Yang was supported by the NSFC under Grant 61379139, 11526215, 11501156, and the Natural Science Research Project of Higher Education of Anhui Province of China under Grant KJ 2015JD18. The work of K. Feng was supported by the NSFC under Grant 11571007 and 11471178.


  1. 1.
    Arasu K.T., Jungnickel D., Ma S.L., Pott A.: Strongly regular Cayley graphs with \(\lambda -\mu =-1\). J. Comb. Theory Ser. A 67(1), 116–125 (1994).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bao J., Ji L., Wei R., Zhang Y.: New existence and nonexistence results for strong external difference families. arXiv:1612.08385v1 (2016).
  3. 3.
    Baumert L.D., Mills W.H., Ward R.L.: Uniform cyclotomy. J. Number Theory 14, 67–82 (1982).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Davis J.A., Huczynska S., Mullen G.L.: Near-complete external difference families. Des. Codes Cryptogr. doi: 10.1007/s10623-016-0275-7 (2016).
  5. 5.
    Huczynska S., Paterson M.B.: Existence and non-existence results for strong external difference families. arXiv:1611.05652v1 (2016).
  6. 6.
    Ma S.L.: Partial difference sets. Discret Math. 52, 75–89 (1984).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ma S.L.: A survey of partial difference sets. Des. Codes Cryptogr. 4(3), 221–261 (1994).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Martin W., Stinson D.R.: Some nonexistence results for strong external difference families using character theory. arXiv:1601.06432 (2016).
  9. 9.
    Paterson M.B., Stinson D.R.: Combinatorial characterizations of algebraic manipulation detection codes involving generalized difference families. Discret Math. 339, 2891–2906 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Polhill J.: Paley partial difference sets in groups of order \(n^4\) and \(qn^2\) for any odd \(n>1\). J. Comb. Theory Ser. A 117, 1027–1036 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Storer T.: Cyclotomy and Difference Sets. Markham Pub. Co, Chicago (1967).zbMATHGoogle Scholar
  12. 12.
    Wen J., Yang M., Feng K.: The \((n,m,k,\lambda )\)-strong external difference family with \(m\ge 5\) exists. arXiv:1612.09495 (2016).

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • Jiejing Wen
    • 1
  • Minghui Yang
    • 2
  • Fangwei Fu
    • 1
  • Keqin Feng
    • 3
  1. 1.Chern Institute of MathematicsNankai UniversityTianjinChina
  2. 2.State Key Laboratory of Information Security, Institute of Information EngineeringChinese Academy of SciencesBeijingChina
  3. 3.Department of Mathematical SciencesTsinghua UniversityBeijingChina

Personalised recommendations