Recent Results on Difference Sets with Classical Parameters

  • Qing Xiang
Part of the NATO Science Series book series (ASIC, volume 542)


We survey recent results on difference sets with classical parameters. In particular, we discuss constructions of cyclic difference sets from hyperovals and related 2-rank results. We also mention a few conjectures on sequences with two-level autocorrelation as well as some recent results on these conjectures.


Cyclic Code Classical Parameter Symmetric Design Permutation Polynomial Binary Weight 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. Antweiler, M. and Bömer, L. (1992) Complex sequences over GF(p m ) with a two-level autocorrelation function and a large linear span, IEEE Trans. Inform. Theory 38, 120–130.MathSciNetzbMATHCrossRefGoogle Scholar
  2. Assmus, E.F.Jr. and Key, J.D. (1992a) Designs and their codes, Cambridge Tracts in Mathematics, 103, Cambridge University Press, Cambridge.Google Scholar
  3. Assmus, E.F.Jr. and Key, J.D. (1992b) Hadamard matrices and their designs: a coding theoretic approach, Trans. Amer. Math. Soc. 330, 269–293.MathSciNetzbMATHCrossRefGoogle Scholar
  4. Bacher, R. (1994) Cyclic difference sets with parameters (511, 255,127), L ‘Enseignement Matheématique 40, 187–192.MathSciNetzbMATHGoogle Scholar
  5. Baumert, L.D. (1971) Cyclic difference sets, Lecture Notes in Mathematics 182, Springer, New York.Google Scholar
  6. Bose, R.C. and Shrikhande, S.S. (1960) On the construction of sets of mutually orthogonal latin squares and the falsity of a conjecture of Euler, Trans. Amer. Math. Soc. 95, 191–209.MathSciNetzbMATHCrossRefGoogle Scholar
  7. Bridges, W.G., Hall, M. and Hayden, J.L. (1981) Codes and designs, J. Combin. Theory (A) 31, 155–174.MathSciNetzbMATHCrossRefGoogle Scholar
  8. Brouwer, A.E. and Wilbrink, H.A. (1995) Block Designs, in F. Buekenhout (ed.), Handbook of Incidence Geometry, Buildings and Foundations, North-Holland, Amsterdam and New York, pp.349–382.CrossRefGoogle Scholar
  9. Chang, A., Golomb, S.W., Gong, G. and Kumar, P.V. (1998) On ideal autocorrelation sequences arising from hyperovals, to appear in Proceedings of the International Conference on Sequences and their Applications, Dec. 14–17, 1998, Singapore.Google Scholar
  10. Chang, A., Gaal, P., Golomb, S.W., Gong, G. and Kumar, P.V. (1998) On a sequence conjectured to have ideal 2-level autocorrelation function, ISIT 1998, Cambridge.Google Scholar
  11. Chang, A., Helleseth, T. and Kumar, P.V. (1998) Further results on a conjectured 2-level autocorrelation sequence, in Proceedings of the Thirty-Sixth Annual Allerton Conference on Communcation, Control and Computing, Sep. 23–25, 1998, pp.598–599.Google Scholar
  12. Cheng, U. (1983) Exhaustive construction of (255, 127, 63) cyclic difference sets, J. Corn-bin. Theory (A) 35, 115–125.zbMATHCrossRefGoogle Scholar
  13. Cherowitzo, W.E. and Storme, L. (1998) α-flocks with oval herds and monomial hyper-ovals, Finite Fields Appl. 4, 185–199.MathSciNetzbMATHCrossRefGoogle Scholar
  14. Chung, H. and No, J.-S. (1999) Linear span of extended sequences and generalized GMW sequences, submitted.Google Scholar
  15. Dillon, J.F. (1999) Multiplicative difference sets via additive characters, manuscipt.Google Scholar
  16. Dobbertin, H. (1999) Kasami power functions, permutation polynomials and cyclic difference sets, this volume.Google Scholar
  17. Dreier, R.B. and Smith, K.W. (1991) Exhaustive determination of (511, 255,127) cyclic difference sets, unpublished.Google Scholar
  18. Evans, R., Hohmann, H.D.L., Krattenthaler, C. and Xiang, Q. (1999) Gauss sums, Jacobi sums and p-ranks of cyclic difference sets, J. Combin. Theory (A), to appear.Google Scholar
  19. Gaal, P. and Golomb, S.W. (1999) Exhaustive determination of (1023, 511, 255) cyclic difference sets, submitted.Google Scholar
  20. Gao, S. and Wei, W.D. (1993) On non-abelian group difference sets, Disc. Math. 112, 93–102.MathSciNetzbMATHCrossRefGoogle Scholar
  21. Glynn, D. (1983) Two new sequences of ovals in finite Desarguesian planes of even order, in Lecture Notes in Mathematics 1036, Springer, Berlin, pp.217–229.Google Scholar
  22. Goethals, J.M. and Delsarte, P. (1968) On a class of majority logic decodable cyclic codes, IEEE Trans. Inform. Theory 14, 182–188.MathSciNetCrossRefGoogle Scholar
  23. Golomb, S.W. (1997) The use of combinatorial structures in communication signal designs, in C. Mitchell (ed.), Applications of Combinatorial Mathematics, 60, pp.59–78.Google Scholar
  24. Gordon, B., Mills, W.H. and Welch, L.R. (1962) Some new difference sets, Caread. J. Math. 14, 614–625.MathSciNetzbMATHGoogle Scholar
  25. Hall, M.Jr. (1956) A survey of difference sets, Proc. Amer. Math. Soc. 7, 975–986.MathSciNetCrossRefGoogle Scholar
  26. Hamada, N. (1973) On the p—rank of the incidence matrix of a balanced or partially balanced incomplete block design and its applications to error-correcting codes, Hiroshima Math. J. 3, 154–226.MathSciNetGoogle Scholar
  27. Hamada, N. and Ohmori, H. (1975) On the BIB-design having the minimum p-rank, J. Combin. Theory (A) 18, 131–140.MathSciNetzbMATHCrossRefGoogle Scholar
  28. Hirschfeld, J.W.P. (1979) Projective Geometries over Finite Fields, Oxford University Press.Google Scholar
  29. Hohmann, H.D.L. and Xiang, Q. (1999) On binary cyclic codes with few weights, preprint.Google Scholar
  30. Jackson, W.-A. (1993) A characterization of Hadamard designs with SL(2, q) acting transitively, Geom. Ded. 46, 197–206.zbMATHCrossRefGoogle Scholar
  31. Jackson, W.-A. and Wild, P.R. (1997) On GMW designs and cyclic Hadamard designs, Designs, Codes and Cryptography 10, 185–191.MathSciNetzbMATHCrossRefGoogle Scholar
  32. Jungnickel, D. (1992) Difference sets, in J. Dinitz, D. R. Stinson (eds.), Contemporary Design Theory, A Collection of Surveys, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley, New York, pp.241–324.Google Scholar
  33. Jungnickel, D. and Pott, A. (1999) Difference sets: an introduction, this volume. Jungnickel, D. and Tonchev, V. (1999) Decompositions of difference sets, submitted.Google Scholar
  34. Lander, E.S. (1983) Symmetric Designs, an Algebraic Approach, Cambridge University Press, Cambridge.CrossRefGoogle Scholar
  35. Lidl, R. and Niederreiter, H. (1997) Finite Fields, 2nd ed., Encyclopedia of Mathematics and Its Applications, vol. 20, Cambridge University Press, Cambridge, 1997.Google Scholar
  36. MacWilliams, F.J. and Mann, H.B. (1968) On the p-rank of the design matrix of a difference set, Inform. Control 12, 474–488.MathSciNetzbMATHCrossRefGoogle Scholar
  37. MacWilliams, F.J. and Sloane, N.J.A. (1977) The Theory of Error-Correcting Codes, North-Holland, Amsterdam.zbMATHGoogle Scholar
  38. Maschietti, A. (1992) Hyperovals and Hadamard designs, J. Geometry 44, 107–116.MathSciNetzbMATHCrossRefGoogle Scholar
  39. Maschietti, A. (1995) On Hadamard designs associated with a hyperoval, J. Geometry 53, 122–130.MathSciNetzbMATHCrossRefGoogle Scholar
  40. Maschietti, A. (1998) Difference sets and hyperovals, Designs, Codes and Crypt. 14, 89–98.MathSciNetzbMATHCrossRefGoogle Scholar
  41. Michael, T.S. (1996) The p-ranks of skew Hadamard designs, J. Combin. Theory (A) 73, 170–171.MathSciNetzbMATHCrossRefGoogle Scholar
  42. No, J.-S., Chung, H. and Yun, M.-S. (1998) Binary pseudorandom sequences of period 2’ - 1 with ideal autocorrelation generated by the polynomial zd + (z + 1)d, IEEE Trans. Inform. Theory 44, 1278–1282.MathSciNetzbMATHCrossRefGoogle Scholar
  43. No, J.-S., Golomb, S.W., Gong, G., Lee, H.-K. and Gaal, P. (1998) Binary pseudorandom sequences of period 2’ - 1 with ideal autocorrelation, IEEE Trans. Inform. Theory 44, 814–817.MathSciNetzbMATHCrossRefGoogle Scholar
  44. No, J.-S., Lee, H.-K., Chung, H., Song, H.-Y. and Yang, K. (1996) Trace representation of Legendre sequences of Mersenne prime length, IEEE Trans. Inform. Theory 42, 2254–2255.MathSciNetzbMATHCrossRefGoogle Scholar
  45. No, J.-S., Yang, K., Chung, H. and Song, H.-Y. (1996) On the construction of binary sequences with ideal autocorrelation property, in Proc. 1996 IEEE International Symposium on Inform. Theory and Its Appl.,(ISITA ‘86), Victoria, B.C., Canada, pp.837–840.Google Scholar
  46. Norwood, T. and Xiang, Q. (1997) On GMW designs and a conjecture of Assmus and Key, J. Combin. Theory (A) 78, 162–168.MathSciNetzbMATHCrossRefGoogle Scholar
  47. Paley, R.E.A.C. (1933) On orthogonal matrices, J. Math. Phys. MIT 12, 311–320.Google Scholar
  48. Pott, A. (1992) A generalization of a construction of Lenz, in Proc. R. C. Bose Memorial Conference on Combin. Math. and Its Applications, (Calcutta, 1988). Sankhÿa Ser. A 54, special issue, pp.315–318.Google Scholar
  49. Pott, A. (1995) Character Theory and Finite Geometry, Lecture Notes in Mathematics 1601, Springer, Berlin.Google Scholar
  50. Scholtz, R.A. and Welch, L.R. (1984) GMW sequences, IEEE Trans. Inform. Theory 30, 548–553.MathSciNetzbMATHCrossRefGoogle Scholar
  51. Segre, B. (1955) Ovals in a finite projective plane, Canad. J. Math. 7, 414–416.MathSciNetzbMATHCrossRefGoogle Scholar
  52. Shrikhande, S.S. and Singh, N.K. (1962) On a method of constructing incomplete block designs, Sankhÿa, Ser. A 24, 25–32.MathSciNetzbMATHGoogle Scholar
  53. Singer, J. (1938) A theorem in finite projective geometry and some applications to number theory, Trans. AMS 43, 377–385.CrossRefGoogle Scholar
  54. Smith, K.J.C. (1969) On the p-rank of the incidence matrix of points and hyperplanes in a finite projective geometry, J. Combin. Theory (A) 7, 122–129.zbMATHCrossRefGoogle Scholar
  55. Stanley, R.P. (1997) Enumerative Combinatorics, Vol. 1, Wadsworth & Brooks/Cole, Pacific Grove, California, 1986; reprinted by Cambridge University Press, Cambridge, 1997.Google Scholar
  56. Turyn, R.J. (1965) Character sums and difference sets, Pacific J. Math. 15, 319–346.MathSciNetzbMATHGoogle Scholar
  57. Xiang, Q. (1998) On balanced binary sequences with two-level autocorrelation functions, IEEE Trans. Inform. Theory 44, 3153–3156.MathSciNetzbMATHCrossRefGoogle Scholar
  58. Yamamoto, K. (1985) On congruences arising from relative Gauss sums, in: Number Theory and Combinatorics, Japan 198.4, World Scientific Publ., pp.423–446.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1999

Authors and Affiliations

  • Qing Xiang
    • 1
  1. 1.Department of Mathematical SciencesUniversity of DelawareNewarkUSA

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