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Cyclic (v; r, s; λ) Difference Families with Two Base Blocks and v ≤ 50

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Abstract

We construct many new cyclic (v; r, s; λ) difference families with v ≥ 2r ≥ 2s ≥ 4 and v ≤ 50. In particular, we construct the difference families with parameters (45; 18, 10; 9), (45; 22, 22; 21), (47; 21, 12; 12), (47; 19, 15; 12), (47; 22, 14; 14), (48; 20, 10; 10), (48; 24, 4; 12), (50; 25, 20; 20), for which the existence question was an open problem. We point out that the (45; 22, 22; 21) difference family gives a balanced incomplete block design (BIBD) with parameters v = 45, b = 90, r = 44, k = 22, and λ = 21, and that the one with parameters (50; 25, 20; 20) gives a pair of binary sequences of length 50 with zero periodic autocorrelation function (the periodic analog of a Golay pair). The new SDSs include nine new D-optimal designs. A normal form for cyclic difference families (with base blocks of arbitrary sizes) is proposed and used effectively in compiling our selective listings in Tables 3–6 of known and new difference families in the above range.

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References

  1. Abel R.J.R.: Forty-three balanced incomplete block designs. J. Combin. Theory Ser. A 65(2), 252–267 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  2. Andres, T.H.: Some Combinatorial Properties of Complementary Sequences. M.Sc. Thesis, University of Manitoba, Winnipeg (1977)

  3. Arasu K.T., Xiang Q.: On the existence of periodic complementary binary sequences. Des. Codes Cryptogr. 2(3), 257–262 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  4. Ashlock D.: Finding designs with genetic algorithms. In: Wallis, W.D. (ed.) Computational and Constructive Design Theory, pp. 49–65. Kluwer Acad. Publ., Dordrecht (1996)

    Google Scholar 

  5. Bömer L., Antweiler M.: Periodic complementary binary sequences. IEEE Trans. Inform. Theory 36(6), 1487–1494 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  6. Borwein P.B., Ferguson R.A.: A complete description of Golay pairs for lengths up to 100. Math. Comp. 73(246), 967–985 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bose R.C.: On the construction of balanced incomplete block designs. Ann. Eugenics 9, 353–399 (1939)

    Article  MathSciNet  Google Scholar 

  8. Chadjiconstantinidis S., Chadjipadelis T., Sotirakoglou K.: Two cyclic supplementary difference sets and optimal designs in linear models. J. Combin. Math. Combin. Comput. 18, 33–56 (1995)

    MATH  MathSciNet  Google Scholar 

  9. Chadjipantelis T., Kounias S.: Supplementary difference sets and D-optimal designs for n ≡ 2 mod 4. Discrete Math. 57(3), 211–216 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  10. Cohn J.H.E.: On determinants with elements ±1, II. Bull. London Math. Soc. 21(1), 36–42 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  11. Colbourn C.J., Dinitz J.H.: Handbook of Combinatorial Designs, 2nd edition. Chapman & Hall/CRC, Boca Raton (2007)

    MATH  Google Scholar 

  12. Ding C.: Two constructions of (v, (v–1)/2, (v–3)/2) difference families. J. Combin. Des. 16(2), 164–171 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  13. Đoković D.Ž.: Survey of cyclic (v; r, s; λ) difference families with v ≤ 50. Facta Univ. Ser. Math. Inform. 12, 1–13 (1997)

    Google Scholar 

  14. Đoković D.Ž.: Note on periodic complementary sets of binary sequences. Des. Codes Cryptogr. 13(3), 251–256 (1998)

    Article  MathSciNet  Google Scholar 

  15. Đoković D.Ž.: Equivalence classes and representatives of Golay sequences. Discrete Math. 189(1-3), 79–93 (1998)

    Article  MathSciNet  Google Scholar 

  16. Đoković, D.Ž.: Cyclic (v; r, s; λ) difference families with two base blocks and v ≤ 50. arXiv:0707.2173v2 (2007)

  17. Ehlich H.: Determinantenabschätzungen für binäre Matrizen. Math. Z. 83, 123–132 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  18. Feng K., Shiue P.J.-S., Xiang Q.: On aperiodic and periodic complementary binary sequences. IEEE Trans. Inform. Theory 45(1), 296–303 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  19. Gysin M.: New D-optimal designs via cyclotomy and generalised cyclotomy. Australas. J. Combin. 15, 247–255 (1997)

    MATH  MathSciNet  Google Scholar 

  20. Gysin M., Seberry J.: An experimental search and new combinatorial designs via a generalisation of cyclotomy. J. Combin. Math. Combin. Comput. 27, 143–160 (1998)

    MATH  MathSciNet  Google Scholar 

  21. Gysin M., Seberry J.: On new families of supplementary difference sets over rings with short orbits. J. Combin. Math. Combin. Comput. 28, 161–186 (1998)

    MATH  MathSciNet  Google Scholar 

  22. Koukouvinos C., Kounias S., Seberry J.: Supplementary difference sets and optimal designs. Discrete Math. 88(1), 49–58 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  23. Kounias S., Koukouvinos C., Nicolaou N., Kakos A.: The non-equivalent circulant Doptimal designs for n ≡ 2 mod 4, n ≤ 54, n = 66. J. Combin. Theory Ser. A 65(1), 26–38 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  24. Kounias S., Koukouvinos C., Nicolaou N., Kakos A.: The non-equivalent circulant D-optimal designs for n = 90. J. Statist. Plann. Inference 53(2), 253–259 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  25. Martínez L., Đoković D.Ž., Vera-López A.: Existence question for difference families and construction of some new families. J. Combin. Des. 12(4), 256–270 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  26. Morales L.B.: Constructing difference families through an optimization approach: six new BIBDs. J. Combin. Des. 8(4), 261–273 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  27. Pott A.: Finite Geometry and Character Theory. Springer-Verlag, Berlin (1995)

    MATH  Google Scholar 

  28. Seberry J., Yamada M.: Hadamard matrices, sequences and block designs. In: Dinitz, J.H., Stinson, D.R. (eds) Contemporary Design Theory: A Collection of Surveys, pp. 431–560. Wiley, New York (1992)

    Google Scholar 

  29. Wallis, J.S.: Some remarks on supplementary difference sets. In: Infinite and Finite Sets, Vol. III, pp. 1503–1526. North-Holland, Amsterdam (1975)

  30. Takeuchi K.: A table of difference sets generating balanced incomplete block designs. Rev. Inst. Internat. Statist. 30, 361–366 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  31. Wilson R.M.: Cyclotomy and difference families in elementary abelian groups. J. Number Theory 4, 17–47 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  32. Yang C.H.: On designs of maximal (+1, −1)-matrices of order n ≡ 2 (mod 4). Math. Comp. 22, 174–180 (1968)

    MATH  MathSciNet  Google Scholar 

  33. Yang C.H.: Maximal binary matrices and sum of two squares. Math. Comp. 30(133), 148–153 (1976)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada

    Dragomir Ž. Đoković

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  1. Dragomir Ž. Đoković
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Correspondence to Dragomir Ž. Đoković.

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The author was supported by an NSERC Discovery Grant.

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Đoković, D.Ž. Cyclic (v; r, s; λ) Difference Families with Two Base Blocks and v ≤ 50. Ann. Comb. 15, 233–254 (2011). https://doi.org/10.1007/s00026-011-0092-7

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