Designs from cyclotomy

  • Elizabeth J. Morgan
  • Anne Penfold Street
  • Jennifer Seberry Wallis
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 560)


In this note we use the theory of cyclotomy to help us construct initial blocks from which we can develop balanced and partially balanced incomplete block designs. Our main construction method, using unions of cyclotomic classes, gives us upper bounds on m, the number of associate classes of the design, but not exact values for m; we discuss the possible values of m and the circumstances under which m=1, so that the design is in fact balanced.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    R.C. Bose and K.R. Nair, Partially balanced incomplete block designs, Sankhya 4 (1938), 337–372.Google Scholar
  2. [2]
    L.E. Dickson, Cyclotomy, higher congruences and Waring’s problem, Amer. J. Math. 57 (1935), 391–424.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    C.G.J. Jacobi, Canon Arithmeticus. (Akademie-Verlag, Berlin, 1956).MATHGoogle Scholar
  4. [4]
    Emma Lehmer, On residue difference sets, Canad. J. Math. 5 (1953), 425–432.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    J.B. Muskat, The cyclotomic numbers of order fourteen, Acta Arith. XI (1966), 263–279.MathSciNetMATHGoogle Scholar
  6. [6]
    Damaraju Raghavarao, Constructions and Combinatorial Problems in Design of Experiments. (John Wiley and Sons Inc., New York, London, Sydney, Toronto, 1972).MATHGoogle Scholar
  7. [7]
    D.A. Sprott, A note on balanced incomplete block designs, Canad. J. Math. 6 (1954), 341–346.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    D.A. Sprott, Some series of partially balanced incomplete block designs, Canad. J. Math. 7 (1955), 369–381.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    Thomas Storer, Cyclotomy and difference Sets. (Markham Publishing Co., Chicago, 1967).MATHGoogle Scholar
  10. [10]
    Anne Penfold Street and W.D. Wallis, Nested designs from sum-free sets, Combinatorial Math. III, Proc. Third Australian Conf., Lecture Notes in Math. 452, 214–226 (Springer-Verlag, Berlin, Heidelberg, New York, 1975).CrossRefGoogle Scholar
  11. [11]
    Jennifer Seberry Wallis, Some remarks on supplementary difference sets, Colloq. Math. Soc. Janos Bolyai 10. Infinite and finite sets, Keszthely, Hungary, 1973, 1503–1526.Google Scholar
  12. [12]
    Jennifer Wallis, A note on BIBDs, J. Austral. Math. Soc., 16 (1973), 257–261.MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    W.D. Wallis, Anne Penfold Street and Jennifer Seberry Wallis, Combinatorics: Room Squares, Sum-Free Sets, Hadamard Matrices. Lecture Notes in Math. 292 (Springer-Verlag, Berlin, Heidelberg, New York, 1972).CrossRefGoogle Scholar
  14. [14]
    A.L. Whiteman, The cyclotomic numbers of order ten, Proc. Symposia App. Math., X (1960), 95–111.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag 1976

Authors and Affiliations

  • Elizabeth J. Morgan
    • 1
    • 2
    • 3
  • Anne Penfold Street
    • 1
    • 2
    • 3
  • Jennifer Seberry Wallis
    • 1
    • 2
    • 3
  1. 1.Department of MathematicsUniversity of QueenslandSt. Lucia
  2. 2.Department of MathematicsUniversity of QueenslandSt. Lucia
  3. 3.Department of Mathematics, Institute of Advanced StudiesAustralian National UniversityCanberra

Personalised recommendations