Designs, Codes and Cryptography

, Volume 40, Issue 2, pp 167–185 | Cite as

Constructions of External Difference Families and Disjoint Difference Families

  • Yanxun Chang
  • Cunsheng Ding


External difference families (EDFs) are a type of new combinatorial designs originated from cryptography. In this paper, some earlier ideas of recursive and cyclotomic constructions of combinatorial designs are extended, and a number of classes of EDFs and disjoint difference families are presented. A link between a subclass of EDFs and a special type of (almost) difference sets is set up.


Difference sets Difference systems of sets Disjoint difference families External difference families 

AMS Classification

05B05 94A66 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arasu, KT, Ding, C, Helleseth, T, Vijay Kumer, P, Martinsen, H 2001Almost difference sets and their sequences with optimal autocorrelationIEEE Trans Inform Theory4728342843Google Scholar
  2. 2.
    Bose, RC 1939On the construction of balanced incomplete block designsAnn Eugen9353399MATHGoogle Scholar
  3. 3.
    Buratti, M 1998Recursive constructions for difference matrices and relative difference familiesJ Combin Des6165182MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Chang, Y, Ji, L 2004Optimal (4up, 5, 1) optical orthogonal codesJ Combin Des5346361MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Chen, K, Zhu, L 1999Existence of (q,k,1) difference families with q a prime power and k = 4,5J Combin Des72130MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Colbourn, MJ, Colbourn, CJ 1984Recursive constructions for cyclic block designsJ Statist Plann Inference1097103MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Colbourn, CJ, Launey, W 1996Difference matricesColbourn, CJDinitz , JH eds. The CRC Handbook of Combinatorial DesignsCRC PressBoca Raton287297Google Scholar
  8. 8.
    Ding C, Yuan J (to appear) A family of skew difference sets. J Comb Theory AGoogle Scholar
  9. 9.
    Dinitz, JH, Rodney, P 1997Disjoint difference families with block size 3Utilitas Math52153160MATHMathSciNetGoogle Scholar
  10. 10.
    Dinitz, JH, Shalaby, N 2002Block disjoint difference families for Steiner triple systems: v≡ 1 (mod 6)J Statist Plann Inference1067786MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Fuji-Hara, R, Miao, Y, Shinohara, S 2002Complete sets of disjoint difference families and their applicationsJ Statist Plann Inference10687103MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Hall, M,Jr 1956A survey of difference setsProc Amer Math Soc6975986CrossRefGoogle Scholar
  13. 13.
    Levenshtein, VI 1971One method of constructing quasi codes providing synchronization in the presence of errorsProb Infor Transm7215222Google Scholar
  14. 14.
    Levenshtein, VI 2004Combinatorial problems motivated by comma-free codesJ Combin Des12184196MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Lidl, R, Niederreiter, H 1983Finite fields. Encyclopedia of mathematics and its applications vol. 20Cambridge University PressCambridgeGoogle Scholar
  16. 16.
    Mutoh Y Difference systems of sets and cyclotomoy II, preprint.Google Scholar
  17. 17.
    Mutoh Y, Tonchev VD (to appear) Difference systems of sets and cyclotomoy. Discrete MathGoogle Scholar
  18. 18.
    Ogata, W, Kurosawa, K, Stinson, DR, Saido, H 2004New combinatorial designs and their applications to authentication codes and secret sharing schemesDiscrete Math279383405MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Paley, REAC 1933On orthogonal matricesJ Math Phys MIT12311320MATHGoogle Scholar
  20. 20.
    Stanton, RG, Mullin, RC 1968Construction of room squaresAnn Math Statist3915401548MATHMathSciNetGoogle Scholar
  21. 21.
    Storer T (1967) Cyclotomy and difference Sets. Markham, ChicagoGoogle Scholar
  22. 22.
    Sze, TW, Chanson, S, Ding, C, Helleseth, T, Parker, MG 2003Logarithm authentication codesInfor Comput14893108MathSciNetCrossRefGoogle Scholar
  23. 23.
    Tonchev, VD 2003Difference systems of sets and code synchronizationRendiconti del Seminario Matematico di Messina Ser II9217226MathSciNetGoogle Scholar
  24. 24.
    Wilson, RM 1972Cyclotomy and difference families in elementary Abelian groupsJ Number Theory41742MATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Yin, J 1998Some combinatorial constructions for optical orthogonal codesDiscrete Math185201219MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2006

Authors and Affiliations

  1. 1.Department of MathematicsBeijing Jiaotong UniversityBeijingChina
  2. 2.Department of Computer ScienceThe Hong Kong University of Science and TechnologyKowloonChina

Personalised recommendations