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On Abelian Difference Sets

  • Chapter
Algebra

Part of the book series: Trends in Mathematics ((TM))

Abstract

We review some existence and nonexistence results — new and old — on abelian difference sets. Recent surveys on difference sets can be found in Arasu (1990), Jungnickel (1992a, b)Pott (1995), Jungnickel and Schmidt (1997), and Davis and Jedwab (1996). Standard references for difference sets are Baumert (1971), Beth et al. (1998), and Lander (1983). This article presents a flavour of the subject, by discussing some selected topics.

Work partially supported by NSA grant # MDA 904-97-1-0012 and by AFOSR grant F49620-96-1-0328.

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References

  • Arasu, K.T., (81, 16, 3) Abelian difference sets do not exist. J. Combin. Th., A43, 350–353, 1986.

    Article  MathSciNet  Google Scholar 

  • Arasu, K.T., Abelian projective planes of square order. Europ. J. Combin., 10, 207–209, 1989.

    MathSciNet  MATH  Google Scholar 

  • Arasu, K.T., More missing entries in Lander’s table could be filled. Arch. Math., 51, 188–192, 1988.

    Article  MathSciNet  MATH  Google Scholar 

  • Arasu, K.T., On abelian difference sets. Arch. Math., 48, 491–494, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  • Arasu, K.T., Recent results on difference sets, In: Coding Theory and Design Theory, Part II(Ed. D. Ray-Chaudhuri) Springer, Heidelberg, 1–23, 1990.

    Google Scholar 

  • Arasu, K.T., Singer groups of biplanes of order 25. Arch. Math., 53, 622–624, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  • Arasu, K.T., On the existence of (400, 57, 8) abelian difference sets, (1997), manuscript.

    Google Scholar 

  • Arasu, K.T., Davis, J.A., Jedwab, J., Ma, S.L., and McFarland, R.L., Exponent bounds for a family of abelian difference sets. In: Groups, Difference Sets, and the Monster. (Eds. K.T. Arasu, J.F. Dillon, K. Harada, S.K. Sehgal and R.L. Solomon) DeGruyter Verlag, Berlin/New York, 129–143, 1996.

    Chapter  Google Scholar 

  • Arasu, K.T., Davis, J.A., Jedwab, J., and Sehgal, S.K., New constructions of Menon difference sets, J. Combin. Th., A64, 329–336, 1993.

    Article  MathSciNet  Google Scholar 

  • Arasu, K.T., Davis, J.A., and Jedwab, J., A nonexistence result for abelian Menon difference sets using perfect binary arrays, Combinatorica, 15(3), 311–317, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  • Arasu, K.T., Davis, J.A., Jungnickel, D., and Pott, A., A note on intersection numbers of difference sets. Europ. J. Combin., 11, 95–98, 1990.

    MathSciNet  MATH  Google Scholar 

  • Arasu, K.T., and Ma, S.L., Abelian difference sets without self-conjugacy. Design, Codes, and Cryptography, 1998 to appear.

    Google Scholar 

  • Arasu, K.T., and Mavron, V.C., Biplanes and singer groups, In: Coding Theory, Design Theory, Group Theory, (Eds. D. Jungnickel, S.A. Vanstone) Wiley, New York, 111–119, 1993.

    Google Scholar 

  • Arasu, K.T., McDonough, T., and Sehgal, S.K., Sums of roots of unity, In: Group Theory, (Eds. R. Solomon and S.K. Sehgal) World Scientific, Singapore, 6–20, 1993.

    Google Scholar 

  • Arasu, K.T., and Ray-Chaudhuri, D.K., Multiplier theorem for a difference list. Ars Combin., 22, 119–137, 1986.

    MathSciNet  MATH  Google Scholar 

  • Arasu, K.T., and Sehgal, S.K., Some new results on abelian difference sets, Utilitas Math., 42, 225–233, 1992.

    MathSciNet  MATH  Google Scholar 

  • Arasu, K.T., and Sehgal, S.K., Difference sets in abelian groups of p-rank two. Designs, Codes, and Cryptography, 5, 5–12, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  • Arasu, K.T., and Sehgal, S.K., Non-existence of some difference sets. JCMCC, 1998 to appear.

    Google Scholar 

  • Arasu, K.T., and Sehgal, S.K., Some new difference sets. J. Combin. Th., A69, 170–172, 1995.

    Article  MathSciNet  Google Scholar 

  • Arasu, K.T., and Xiang, Q., Multiplier theorems. J. Comb. Des., 3, 257–267, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  • Baumert, L.D., (1977), Cyclic difference sets. Lect. Not. Math., Springer, New York, 182, 1971.

    Google Scholar 

  • Baumert, L.D., and Fredricksen, H., The cyclotomic numbers of order 18 with applications to difference sets. Math. Comp., 21, 204–219, 1967.

    Article  MathSciNet  MATH  Google Scholar 

  • Baumert, L.D., Mills, W.H., and Ward, R.L., Uniform cyclotomy. J. Nr. Th., 14, 67–82, 1982.

    MathSciNet  MATH  Google Scholar 

  • Beth, T., Jungnickel, D., and Lenz, H., Design Theory, Cambridge Univ. Press, Cambridge (2ndedition), 1998.

    Google Scholar 

  • Bozikov, Z., Abelian Singer groups of certain symmetric block designs. Radovi Mat., 1, 247–253, 1985.

    MathSciNet  MATH  Google Scholar 

  • Broughton, W.J., A note on table 1 of “Barker sequences and difference sets”. L’Ens. Math., 50, 105–107, 1994.

    MathSciNet  Google Scholar 

  • Brualdi, R.A., A note on multipliers of difference sets. J Res. Nat Bur. Standards B, 69, 87–89, 1965.

    MathSciNet  MATH  Google Scholar 

  • Bruck, R.H., Difference sets in a finite group. Trans. Amer. Math. Soc., 78, 464–481, 1955.

    Article  MathSciNet  MATH  Google Scholar 

  • Bruck, R.H., and Ryser, H.J., The non-existence of certain finite projective planes. Canad. J. Math., 1, 88–93, 1949.

    Article  MathSciNet  MATH  Google Scholar 

  • Camion, R., and Mann, H.B., Antisymmetric difference sets. J. Nr. Th., 4, 266–268; 1972.

    MathSciNet  MATH  Google Scholar 

  • Chan, W.K., Necessary conditions for Menon difference sets, Design, Codes, and Cryptography, 3, 147–154, 1993.

    Article  MATH  Google Scholar 

  • Chan, W.K., and Siu, M.K., Summary of perfects s× tarrays, 1 ≤ s ≤ t ≤100, Electr. Lett., 27, 709,710, 1991.

    Google Scholar 

  • Chan, W.K., and Siu, M.K., Correction to “summary of perfect s x tarrays, 1 ≤ s ≤ t ≤100”, Electr. Letter, 27, 1112, 1991.

    Article  Google Scholar 

  • Chan, W.K., Siu, M.K., and Ma, S.L., Nonexistence of certain perfect arrays, Discrete Math., 125, 107–113, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  • Chen, Y.Q., On the existence of abelian Hadamard difference sets and a new family of difference sets. Finite Fields Appl., 3, 234–256, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  • Chen Y.Q., Xiang, Q., and Sehgal, S.K., An exponent bound on skew Hadamard abelian difference sets. Designs, Codes, and Cryptography, 4, 313–317, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  • Chowla, S., A property of biquadratic residues. Proc Nat. Acad. Sci. India, A14, 45–46, 1994.

    MathSciNet  Google Scholar 

  • Chowla, S., and Ryser, HJ., Combinatorial problems. Canad. J. Math., 2, 93–99, 1950.

    Article  MathSciNet  MATH  Google Scholar 

  • Cohen, S.D., Generators in cyclic difference sets. J. Combin. Th., A51, 227–236, 1989.

    Article  Google Scholar 

  • Colbourn, C.J., and Dinitz, J.H., (Eds.), The CRC Handbook of Combinatorial Designs, CRC Press, Boca Raton, 1996.

    MATH  Google Scholar 

  • Davis, J., A generalization of Kraemer’s result on difference sets. J. Combin. Th., A59, 187–192, 1992.

    Article  Google Scholar 

  • Davis, J., A result on Dillon’s conjecture in difference sets. J. Combin. Th., A57, 238–242, 1991.

    Article  Google Scholar 

  • Davis, J.A., Difference sets in abelian 2-groups, J. Combin. Th., A57, 262–286, 1991.

    Article  Google Scholar 

  • Davis, J.A., and Jedwab, J., A Survey of Hadamard difference sets, In: Groups, Difference Sets, and the Monster, (Eds. K.T. Arasu, J. Dillon, K. Haradu, S.K. Sehgal and R. Solomon), deGruyter Verlag, Berlin/New York, 145–156, 1996.

    Chapter  Google Scholar 

  • Davis, J. A., and Jedwab, J., A unifying construction of difference sets. J. Combin. Th., A80, 13–78, 1997.

    Article  MathSciNet  Google Scholar 

  • Davis, J.A., and Jedwab, J., A summary of Menon difference sets. Congr. Nr., 93, 203–207, 1993.

    MathSciNet  MATH  Google Scholar 

  • Davis, J.A., and Jedwab, J., Some recent developments in difference sets. In: Combinatorical Designs and Applications (Eds. K. Quinn et al.,)Addison-Wesley, London, (to appear).

    Google Scholar 

  • Davis, J.A., and Jedwab, J., Nested Hadamard difference sets, J. Stat. Planning & Inference, 62, 13–20, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  • Dillon, J.F., A Survey of difference sets in 2-groups. Presented at the Marshall Hall Memorial Conference, Burlington, VT, 1990a.

    Google Scholar 

  • Dillon, J.F., Cyclic difference sets and primitive polynomials. In: Finite Fields, Coding Theory and Advances in Communications and Computing (Eds:G.L. Mullen and P.J.S. Shiue), Marcel Dekker, New York, 436–437, 1993.

    Google Scholar 

  • Dillon, J.F., Difference sets in 2-groups. In: Proc. NSA Math. Sci. Meet., Ft. George Meade, Maryland, (Ed. R.L. Ward) 165–172, 1987.

    Google Scholar 

  • Dillon, J.F., Difference sets in 2-groups, In: Finite Geometries and Combinatorial Designs, Contemporary Mathematics. Vol. III (Ed. E.S. Kraemer), Birkhäuser, Boston, 65–72, 1990a.

    Chapter  Google Scholar 

  • Dillon, J.F., Elementary Hadamard difference sets. Ph.D. Thesis, University of Maryland, 1974.

    Google Scholar 

  • Dillon, J.F., Elementary Hadamard difference sets. In: Proc. Sixth Southeastern Conf. On Combinatorics, Graph Theory and Computing, 237–249, 1975.

    Google Scholar 

  • Dillon, J.F., Variations on a scheme of McFarland for noncyclic difference sets. J. Combin. Th., A40, 9–21, 1985.

    Article  MathSciNet  Google Scholar 

  • Eckmann, A., Eliahou, S., and Kervaire, M., Compressible difference lists. J. Statist. Planning & Inf., 62(1), 35–38, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  • Eliahou, S., and Kervaire, M., A note on the equation θθ̄ = n +λΣ.J. Statist. Planning & Inf., 62(1), 21–34, 1997.

    Article  MathSciNet  Google Scholar 

  • Eliahou, S., and Kervaire, M., Barker sequences and difference sets. L’Ens. Math., 38, 345–382, 1992.

    MathSciNet  MATH  Google Scholar 

  • van Eupen, M., and Tonchev, V.D., Linear codes and the existence of a reversible Hadamard difference set in Z 2 ×Z 2 × Z 4 5 , J. Combin. Th., A79, 161–167, 1997.

    Google Scholar 

  • Golomb, S.W., and Song, H.Y., A conjecture on the existence of cyclic Hadamard difference sets. J. Statist. Planning Inf., 62(1), 39–42, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  • Gordon, B., Mills, W.H., and Welsch, L.R., Some new difference sets. Canad. J. Math., 14, 614–625, 1962.

    Article  MathSciNet  MATH  Google Scholar 

  • Gordon, D.M., The prime power conjecture is true for n <2,000,000. Electron. J. Comb., 1, 1994.

    Google Scholar 

  • Hall, M., Jr., Cyclic projective planes. Duke J. Math., 14, 1079–1090, 1947.

    Article  MATH  Google Scholar 

  • Hall, M., Jr., A survey of difference sets. Proc. Amer. Math. Soc, 7, 975–986, 1956.

    Article  MathSciNet  Google Scholar 

  • Hall, M., Jr., Combinatorial Theory. 2nd Ed. Wiley, New York, 1986.

    MATH  Google Scholar 

  • Hall, M. Jr., and Ryser, H.J., Cyclic incidence matricess. Canad. J. Math., 3, 495–502, 1951.

    Article  MathSciNet  MATH  Google Scholar 

  • Hayashi, H.S., Computer investigation of difference sets. Math. Comp., 19, 73–78, 1965.

    Article  MathSciNet  MATH  Google Scholar 

  • Ho, C.Y., On bounds for groups of multipliers of planar difference sets. J. Algebra, 148, 325–336, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  • Ho, C.Y., On multiplier groups of finite cyclic planes. J. Algebra, 122, 250–259, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  • Ho, C.Y., Planar Singer groups with even order multiplier groups. In: Finite Geometry and Combinatorics. (Eds. A. Beutelspacher, et al.,) Cambridge University Press, Cambridge, 187–198, 1993.

    Chapter  Google Scholar 

  • Ho, C.Y., Projective planes with a regular collineation group and a question about powers of aprime. J. Algebra, 154, 141–151, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  • Ho, C.Y., Singer groups, an approach from a group of multipliers of even order. Proc. Amer. Math. Soc, 119, 925–930, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  • Ho, C.Y., Some basic properties of planar Singer groups. Geom. Ded., 55, 59–70, 1995.

    Article  MATH  Google Scholar 

  • Ho, C.Y., and Pott, A., On multiplier groups of planar difference sets and a theorem of Kantor. Proc. Amer. Math. Soc, 109, 803–808, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  • Iiams, J., Liebler, R., and Smith, K.W., In: Groups, Difference sets and the monster, (Eds: K.T. Arasu et al.,), DeGruyter Verlag, Berlin/New York, 157–168, 1996.

    Chapter  Google Scholar 

  • Ireland, K., and Rosen, M., A Classical Introduction to Modern Number Theory. Springer, New York, 1982.

    MATH  Google Scholar 

  • Jedwab, J., Generalized perfect arrays and Menon difference sets, Designs, Codes, and Cryptography, 2, 19–68, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  • Jedwab, J., Non-existence of perfect binary arrays. Electron. Lett., 27, 1252–1254, 1991.

    Article  MATH  Google Scholar 

  • Jedwab, J., Non-existence of perfect binary arrays. Electron. Lett., 29, 99–101, 1993.

    Article  Google Scholar 

  • Jedwab, J., Mitchell, C., Piper, F., and Wild, P., Perfect binary arrays and difference sets. Discr. Math., 125, 241–254, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  • Johnsen, E.C., Skew-Hadamard abelian group difference sets. J. Algebra, 4, 388–402, 1966.

    Article  MathSciNet  MATH  Google Scholar 

  • Jungnickel, D., Difference sets, In: Comtemporary Design Theory: A Collection of Surveys (Eds. J.H. Dinitz and D.R. Stinson), Wiley, New York, 241–324, 1992a.

    Google Scholar 

  • Jungnickel, D., On Lander’s multiplier theorem for difference lists. J. Comb. Inf. System Sci., 17, 123–129, 1992b.

    MathSciNet  MATH  Google Scholar 

  • Jungnickel, D., and Pott, A., Difference sets: abelian. In: The CRC Handbook of Combinatorial Designs (Eds. C.J. Colbourn and J. Dinitz) CRC Press, Boca Raton, 297–312, 1996.

    Google Scholar 

  • Jungnickel, D., and Pott, A., Two results on difference sets. Coll. Math. Soc. János Bolyai, 52, 325–330, 1988.

    MathSciNet  Google Scholar 

  • Jungnickel, D., and Schmidt, B., Difference sets: an update, In: Combinatorial Designs and Related Structure (Eds. J.W.P. Hirschfeld, S.S. Magliveras, and M.J. de Resmini), London Math. Soc. Lect. Not., 245, 89–112, 1997.

    Chapter  Google Scholar 

  • Jungnickel, D., and Vedder, K., On the geometry of planar difference sets. Europ. J. Combin., 5, 143–148, 1984.

    MathSciNet  MATH  Google Scholar 

  • Kantor, W.M., Exponential numbers of two-weight codes, difference sets and symmetric designs. Discrete Math., 46, 95–98, 1984.

    Article  MathSciNet  Google Scholar 

  • Kantor, W.M., Primitive permutation groups of odd degree, and an application to finite projective planes. J. Alg., 106, 15–45, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  • Kibler, R.E., A summary of non-cyclic difference sets, k <20. J. Combin. Th., A25, 62–67, 1978.

    Article  MathSciNet  Google Scholar 

  • Kopilovich, L.E., Difference sets in non-cyclic abelian groups. Kibernetika, 2, 20–23, 1989.

    MathSciNet  Google Scholar 

  • Kraemer, R.G., Proof of a conjecture on Hadamard 2-groups, J. Combin. Th., A63, 1–10, 1993.

    Article  MathSciNet  Google Scholar 

  • Lander, E.S., Symmetric Designs: An Algebraic Approach, Cambridge University Press, Cambridge, 1983.

    Book  MATH  Google Scholar 

  • Lander, E.S., Restrictions upon multipliers of an abelian difference set. Arch. Math., 50, 241–242, 1988.

    Article  MathSciNet  MATH  Google Scholar 

  • Lehmer, E., On residue difference sets. Canad. J. Math., 5, 425–432, 1953.

    Article  MathSciNet  MATH  Google Scholar 

  • Ma, S.L., and Schmidt, B., A sharp exponent bound for McFarland difference sets with p= 2. J. Comb. Th., A80, 347–352, 1997a.

    MathSciNet  Google Scholar 

  • Ma, S.L., and Schmidt, B:, Difference sets corresponding to a class of symmetric designs. Designs, Codes and Cryptography, 10, 223–236, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  • Ma, S.L., and Schmidt, B., The structure of abelian groups containing McFarland difference sets, J. Comb. Th., A70, 313–322, 1995.

    MathSciNet  Google Scholar 

  • Ma, S.L., and Schmidt, B., On (p a , p, p a , p a-1-relative difference sets. Designs, Codes and Cryptography, 6, 57–71, 1995a.

    Article  MathSciNet  MATH  Google Scholar 

  • Mann, H.B., Some theorems on difference sets. Canad. J. Math., 4, 222–226, 1952.

    Article  MATH  Google Scholar 

  • Mann, H.B., Balanced incomplete block designs and abelian difference sets. Illinois J. Math., 8, 252–261, 1964.

    MathSciNet  MATH  Google Scholar 

  • Mann, H.B., Addition Theorems, Wiley, New York, 1965.

    MATH  Google Scholar 

  • Mann, H.B., Difference sets in elementary abelian groups. Illinois J. Math., 9, 212–219, 1965.

    MathSciNet  MATH  Google Scholar 

  • Mann, H.B., and McFarland, R.L., On Hadamard difference sets, In: A Survey of Combinatorial Theory(Eds. J.N. Srivastava et al.), North-Holland, Amsterdam, 333–334, 1973.

    Google Scholar 

  • Mann, H.B., and Zaremba, S.K., On multipliers of difference sets. Illinois J. Math., 13, 378–382, 1969.

    MathSciNet  MATH  Google Scholar 

  • McFarland, R.L., Difference sets in abelian groups of order 4p 2, Mitt. Math. Sem., Giessen, 192,1–70, 1989.

    MathSciNet  Google Scholar 

  • McFarland, R.L., A family of difference sets in non-cyclic abelian groups. J. Comb. Th., A15, 1–10, 1973.

    MathSciNet  Google Scholar 

  • McFarland, R.L., On multipliers of abelian difference sets, Ph. D. Thesis., Ohio State University, 1970.

    Google Scholar 

  • McFarland, R.L., and Mann, H.B., On multipliers of abelian difference sets. Canad. J. Math., 17, 541–542, 1965.

    Article  MathSciNet  MATH  Google Scholar 

  • McFarland, R.L., Necessary conditions for Hadamard difference sets, In: Coding Theory and Design Theory(Ed. D.K. Ray-Chaudhuri), Springer, New York, 257–272, 1990a.

    Google Scholar 

  • McFarland, R.L., Subdifference sets of Hadamard difference sets, J. Combin. Th., A54, 112–122, 1990b.

    Article  MathSciNet  Google Scholar 

  • McFarland, R.L., and Rice, B.F., Translates and multipliers of abelian difference sets. Proc. Amer. Math. Soc., 68, 375–379, 1978.

    Article  MathSciNet  MATH  Google Scholar 

  • Menon, P.K., Difference sets in abelian groups. Proc. Amer. Math. Soc, 11,368–376, 1960.

    Article  MathSciNet  MATH  Google Scholar 

  • Menon, P.K., On difference sets whose parameters satisfy a certain relation, Proc. Amer. Math. Soc, 13,739–745, 1962.

    Article  MathSciNet  MATH  Google Scholar 

  • Muzychuk, M., Difference sets with n= 2p m . J. algeb. Combin., 7, 77–89, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  • Newman, M., Multipliers of difference sets. Canad. J. Math., 15, 121–124, 1963.

    Article  MathSciNet  MATH  Google Scholar 

  • Ostrom, T.G., Concerning difference sets. Canad. J. Math., 5, 421–424, 1953.

    Article  MathSciNet  MATH  Google Scholar 

  • Pott, A., Applications of the DFT to abelian difference sets. Arch. Math., 51, 283–288, 1988.

    Article  MathSciNet  MATH  Google Scholar 

  • Pott, A., Differenzenmengen und Gruppenalgebren, Ph.D. Thesis, Univ. Giessen, 1988.

    Google Scholar 

  • Pott, A., Finite geometry and character theory, Lect. Not. Math., 1601, Springer-Verlag, Berlin, 1995.

    MATH  Google Scholar 

  • Pott, A., New necessary conditions on the existence of abelian difference sets. Combinatorica, 12, 89–93, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  • Pott, A., On abelian difference set codes. Designs, Codes and Crypt., 2, 263–271, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  • Pott, A., On multipliers theorems. In: Coding Theory and Design Theory, Part II (Ed. D. Ray-Chaudhuri). Springer, New York, 286–289, 1990.

    Google Scholar 

  • Qiu, W., A method for studying the multiplier conjecture and a partial solution to it. Ars. Comb., 39, 5–23, 1995.

    MATH  Google Scholar 

  • Qiu, W., A necessary condition on the existence of abelian difference sets. Discr. Math., 137, 383–386, 1995.

    Article  MATH  Google Scholar 

  • Qiu, W., Further results on the multiplier conjecture for the case n= 2n 1 . J. Comb. Math. Comb. Comp., 20, 27–31, 1996.

    MATH  Google Scholar 

  • Qiu, W., On the multiplier conjecture. Acta. Math. Sinica, New Series, 10, 49–58, 1994.

    Article  MATH  Google Scholar 

  • Qiu, W., Proving the multiplier theorem using representation theory of groups. Northeast Math. J., 9, 169–172, 1993.

    MathSciNet  MATH  Google Scholar 

  • Qiu, W., The multiplier conjecture for elemenatary abelian groups. J. Comb. Des., 2, 117–129, 1994.

    Article  MATH  Google Scholar 

  • Qiu, W., The multiplier conjecture for the case n = 4n 1 .J. Comb. Des., 3, 393–397, 1995.

    Article  MATH  Google Scholar 

  • Ray-Chaudhuri, D.K., and Xiang, Q., New necessary conditions for abelian Hadamard difference sets, J. Stat. Planning & Inf., 62, 69–80, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  • Rothaus, O.S., On “bent” functions. J. Combin. Th., A20, 300–305, 1976.

    Article  MathSciNet  Google Scholar 

  • Schmidt, B., Circulant Hadamard matrices: overcoming non-selfconjugacy. Submitted.

    Google Scholar 

  • Schmidt, B., Cyclotomic integers of prescribed absolute value and the class group. Submitted.

    Google Scholar 

  • Schmidt, B., “Defferenzmengen und relative Defferenzmengen”, Augsburger Mathematisch-Naturwissenschaftliche Schriften Band 4, Augsburg, 1995.

    Google Scholar 

  • Schmidt, B., Non-existence results on Chen and Davis-Jedwab difference sets. Submitted.

    Google Scholar 

  • Schutzenberger, M.P., A non-existence theorem for an infinite family of symmetrical block designs. Ann. Eugenics., 14, 186–187, 1949.

    MathSciNet  Google Scholar 

  • Singer, J., A theorem in finite projective geometry and some applications to number theory. Trans. Amer, math Soc., 43, 377–385, 1938.

    Article  MathSciNet  Google Scholar 

  • Smith, K.W., A table on non-abelian difference sets. In: “CRC Handbook of Combinatorial Designs” (Eds. C.J. Colbourn and J.H. Dinitz), CRC press, Boca Raton, 308–312, 1996.

    Google Scholar 

  • Song, H.Y., and Golomb, S.W., On the existence of cyclic Hadamard difference sets. IEEE Trans. Inf. Th., 40, 1266–1268, 1994.

    Article  MATH  Google Scholar 

  • Spence, T., A family of difference sets in non-cyclic groups. J. Combin. Th., A22, 103–106, 1977.

    Article  MathSciNet  Google Scholar 

  • Stanton, R.G., and Sprott, DA, A family of difference sets. Canad. J. Math., 11, 73–77, 1958.

    Article  MathSciNet  Google Scholar 

  • Storer, J., Cyclotomy and Difference Sets. Markham, Chicago, 1967.

    MATH  Google Scholar 

  • Takeuchi, K., A table of difference sets generating balanced incomplete block designs. Rev. Inst. int. Statist., 30, 361–366, 1962.

    Article  MATH  Google Scholar 

  • Turyn, R.J., Character sums and difference sets, Pacific J. Math., 15, 319–346, 1965.

    MathSciNet  MATH  Google Scholar 

  • Turyn, R.J., A special class of Willaimson matrices and difference sets, J. Combin. Th., A36, 111–115, 1984.

    Article  MathSciNet  Google Scholar 

  • Turyn, R.J., The multiplier theorem for difference sets. Canad. J. Math., 16, 386–388, 1964.

    Article  MathSciNet  MATH  Google Scholar 

  • van Eupen, M., and Tonchev, V.D., Linear codes and the existence of reversible Hadamard difference sets in Z 2 ×Z 2 × Z 45 , J. Combin. Th., A79, 161–167, 1997.

    Article  Google Scholar 

  • Vera Lopez, A., and Garcia Sanchez, M.A., On the existence of abelian difference sets with 100 < k ≤150. J. Comb. Math. Com. Comp., 23, 97–112, 1997.

    MathSciNet  MATH  Google Scholar 

  • Wei, R., Non-existence of some abelian difference sets. In: Combinatorial Design and Applications (Eds. W.D. Wallis, H. Shen, W Wei, and L. Zhu). Marcel Dekker, New York, 159–164, 1990.

    Google Scholar 

  • Whiteman, A.L., A family of difference sets. Illinois J. Math., 6, 107–121, 1962.

    MathSciNet  MATH  Google Scholar 

  • Wilbrink, H.A., A note on planar difference sets. J. Combin. Th., A38, 94–95, 1985.

    Article  MathSciNet  Google Scholar 

  • Wilson, R.M., and Xiang, Q., Constructions of Hadamard difference sets. J. Combin. Th., A77, 148–160, 1997.

    Article  MathSciNet  Google Scholar 

  • Xia, M.Y., Some infinite classes of special Willaimson matrices and difference sets, J. Combin. Th., A36, 111–115, 1992.

    Google Scholar 

  • Xiang, Q., Some results on multipliers and numerical multiplier groups of difference sets. Graph. Combin., 10, 293–304, 1994.

    Article  MATH  Google Scholar 

  • Xiang, Q., and Chen, Y.Q., On the size of the multiplier groups of cyclic difference sets. J. Combin. Th., A69, 168–169, 1995.

    Article  MathSciNet  Google Scholar 

  • Xiang, Q., and Chen, Y.Q., On Xia’s construction of Hadamard difference sets. Finite Field. Appl., 2, 86–95, 1996.

    MathSciNet  Google Scholar 

  • Yamamoto, K., Decomposition fields of difference sets. Pacific J. Math., 13, 337–352, 1953.

    Google Scholar 

  • Yamamoto, K., On Jacobi sums and difference sets. J. Combin. Th., 3, 146–181, 1967.

    Article  MATH  Google Scholar 

  • Yamamoto, K., On the application of half-norms to cyclic difference sets. In: Combinatorial Mathematics and Its Applications (Eds. R.C. Bose and TA. Dowling). University of North Carolina Press, Chapel Hill, 1969.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Wright State University, Dayton, Ohio, 45435, USA

    K. T. Arasu

  2. Department of Mathematics, Ohio State University, Columbus, Ohio, 43210, USA

    Surinder K. Sehgal

Authors
  1. K. T. Arasu
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  2. Surinder K. Sehgal
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Editors and Affiliations

  1. Centre for Advanced Study in Mathematics, Punjab University, Chandigarh, India

    I. B. S. Passi

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© 1999 Hindustan Book Agency (India) and Indian National Science Academy

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Arasu, K.T., Sehgal, S.K. (1999). On Abelian Difference Sets. In: Passi, I.B.S. (eds) Algebra. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9996-3_1

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  • DOI: https://doi.org/10.1007/978-3-0348-9996-3_1

  • Publisher Name: Birkhäuser Basel

  • Print ISBN: 978-3-0348-9998-7

  • Online ISBN: 978-3-0348-9996-3

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