# zbMATH — the first resource for mathematics

A unifying construction for difference sets. (English) Zbl 0884.05019
The paper under review clearly ranks as one of the most important papers in the general area of difference sets ever written. I will just mention the most striking results that make it important: (1) It contains the first new parameter family of difference sets found since 1977. (2) It gives a unifying theory which allows to produce a difference set with $$(v, n)>1$$ in every abelian group which is known to contain such a difference set. Not only was this true at the time of writing the paper, but it also holds for the new families of difference sets which were subsequently discovered by Yuqing Chen [“On the existence of abelian Hadamard difference sets and a new family of difference sets”, Finite Fields Appl. 3, 234-256 (1997); and “A construction of difference sets”, Des. Codes Cryptography 13, 247-250 (1998)]. (3) It characterizes a certain class of groups containing McFarland difference sets. (4) It allows a transparent treatment of the family of Hadamard difference sets, whereas the best previous description (in terms of binary arrays and binary supplementary quadruples) was still rather cumbersome. In particular, the celebrated characterization theorem for those abelian 2-groups which contain a difference set (a problem which took decades to settle) becomes a trivial corollary. (5) It gives a unified way of constructing semiregular relative difference sets in virtually all the known cases, and it gives many new examples.

##### MSC:
 05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.) 51E30 Other finite incidence structures (geometric aspects)
Full Text:
##### References:
 [1] Alquaddoomi, S.; Scholtz, R.A., On the nonexistence of barker arrays and related matters, IEEE trans. inform. theory, 35, 1048-1057, (1989) [2] Arasu, K.T.; Davis, J.A.; Jedwab, J.; Sehgal, S.K., New constructions of menon difference sets, J. combin. theory ser. A, 64, 329-336, (1993) · Zbl 0795.05032 [3] Arasu, K.T.; Sehgal, S.K., Some new difference sets, J. combin. theory ser. A, 69, 170-172, (1995) · Zbl 0822.05004 [4] Beth, T.; Jungnickel, D.; Lenz, H., Design theory, (1986), Cambridge University Press Cambridge [5] Y. Q. Chen, On the existence of abelian Hadamard difference sets and a new family of difference sets, Finite Fields Appl. [6] Y. Q. Chen, A construction of difference sets, Designs, Codes, Cryptogr. · Zbl 0898.05005 [7] Chen, Y.Q.; Ray-Chaudhuri, D.K.; Xiang, Q., Constructions of partial difference sets and relative difference sets using Galois rings II, J. combin. theory ser. A, 76, 179-196, (1996) · Zbl 0859.05021 [8] Davis, J.A., A note on products of relative difference sets, Designs, codes, cryptogr., 1, 117-119, (1991) · Zbl 0754.05014 [9] Davis, J.A., A result on Dillon’s conjecture in difference sets, J. combin. theory ser. A, 57, 238-242, (1991) · Zbl 0756.05031 [10] Davis, J.A., Construction of relative difference sets inp, Discrete math., 103, 7-15, (1992) · Zbl 0777.05023 [11] Davis, J.A., An exponent bound for relative difference sets inp, Ars combin., 34, 318-320, (1992) · Zbl 0776.05019 [12] Davis, J.A.; Jedwab, J., Recursive construction for difference sets, Bull. inst. combin. appl., 13, 128, (1995) · Zbl 0819.05015 [13] Davis, J.A.; Jedwab, J., A survey of Hadamard difference sets, (), 145-156 · Zbl 0847.05016 [14] J. A. Davis, J. Jedwab, Some recent developments in difference sets, Combinatorial Designs and Their Applications, K. Quinn, Pitman Research Notes in Mathematics, Addison-Wesley-Longman, London · Zbl 0923.05012 [15] J. A. Davis, J. Jedwab, M. Mowbray, New families of semi-regular relative difference sets, Designs, Codes and Cryptogr. · Zbl 0888.05008 [16] Davis, J.A.; Sehgal, S.K., Using the simplex code to construct relative difference sets in 2-groups, Designs, codes, and cryptogr., 11, 267-277, (1997) · Zbl 0880.05020 [17] Davis, J.A.; Smith, K.W., A construction of difference sets in high exponent 2-groups using representation theory, J. algebraic combin., 3, 137-151, (1994) · Zbl 0797.05018 [18] J. F. Dillon, A survey of difference sets in 2-groups · Zbl 0707.05011 [19] J. F. Dillon, 1974, Elementary Hadamard Difference Sets, University of Maryland [20] Dillon, J.F., Variations on a scheme of mcfarland for noncyclic difference sets, J. combin. theory ser. A, 40, 9-21, (1985) · Zbl 0583.05016 [21] Elliott, J.E.H.; Butson, A.T., Relative difference sets, Illinois J. math., 10, 517-531, (1966) · Zbl 0145.01503 [22] van Eupen, M.; Tonchev, V.D., Linear codes and the existence of a reversible Hadamard difference set in $$Z$$_2245, J. combin. theory ser. A, 79, 161-167, (1997) · Zbl 0883.05024 [23] Fenimore, E.E.; Cannon, T.M., Coded aperture imaging with uniformly redundant arrays, Appl. optics, 17, 337-347, (1978) [24] Ganley, M.J., On a paper of Dembowski and Ostrom, Arch. math., 27, 93-98, (1976) · Zbl 0323.20046 [25] Hershey, J.E.; Yarlagadda, R., Two-dimensional synchronisation, Electron. lett., 19, 801-803, (1983) [26] Hoffman, A.J., Cyclic affine planes, Canad. J. math., 4, 295-301, (1952) · Zbl 0048.13101 [27] Jedwab, J., Generalized perfect arrays and menon difference sets, Designs, codes and cryptogr., 2, 19-68, (1992) · Zbl 0767.05030 [28] Jungnickel, D., On automorphism groups of divisible designs, Canad. J. math., 34, 257-297, (1982) · Zbl 0465.05011 [29] Jungnickel, D., Difference sets, (), 241-324 · Zbl 0768.05013 [30] D. Jungnickel, B. Schmidt, Difference sets: An update, Geometry, Combinatorial Designs and Related Structures, J. W. P. HirschfeldS. S. MagliverasM. J. de Resmini · Zbl 0883.05023 [31] Kraemer, R.G., Proof of a conjecture on Hadamard 2-groups, J. combin. theory ser. A, 63, 1-10, (1993) · Zbl 0771.05020 [32] Lander, E.S., Symmetric designs: an algebraic approach, London math. society lecture notes, 74, (1983), Cambridge University Press Cambridge [33] Leung, K.H.; Ma, S.L., Constructions of partial difference sets and relative difference sets onp, Bull. London math. soc., 22, 533-539, (1990) · Zbl 0689.05016 [34] Liebler, R.A., The inversion formula, J. combin. math. combin. comput., 13, 143-160, (1993) · Zbl 0779.05007 [35] Liebler, R.A.; Smith, K.W., On difference sets in certain 2-groups, (), 195-212 [36] Ma, S.L.; Pott, A., Relative difference sets, planar functions and generalized Hadamard matrices, J. algebra, 175, 505-525, (1995) · Zbl 0830.05012 [37] Ma, S.L.; Schmidt, B., On (p^a,ppa,pa)-relative difference sets, Designs, codes and cryptogr., 6, 57-71, (1995) · Zbl 0853.05017 [38] S. L. Ma, B. Schmidt, 1995, A Sharp Exponent Bound for McFarland Difference Sets withp, National University of Singapore [39] Ma, S.L.; Schmidt, B., The structure of the abelian groups containing mcfarland difference sets, J. combin. theory ser. A, 70, 313-322, (1995) · Zbl 0830.05013 [40] Ma, S.L.; Schmidt, B., Difference sets corresponding to a class of symmetric designs, Designs, codes and cryptogr., 10, 223-236, (1997) · Zbl 0869.05014 [41] Martin, S.J.; Butler, M.A.; Land, C.E., Ferroelectric optical image comparator using PLZT thin films, Electron. lett., 24, 1486-1487, (1988) [42] McFarland, R.L., A family of difference sets in non-cyclic groups, J. combin. theory ser. A, 15, 1-10, (1973) · Zbl 0268.05011 [43] D. B. Meisner, 1991, Menon Designs and Related Difference Sets, University of London [44] Meisner, D.B., Families of menon difference sets, Ann. discrete math., 52, 365-380, (1992) · Zbl 0772.05014 [45] Meisner, D.B., New classes of groups containing menon difference sets, Designs, codes and cryptogr., 8, 319-325, (1996) · Zbl 0858.05020 [46] Meisner, D.B., A difference set construction of turyn adapted to semi-direct products, (), 169-174 · Zbl 0844.05024 [47] Pott, A., On the structure of abelian groups admitting divisible difference sets, J. combin. theory ser. A, 65, 202-213, (1994) · Zbl 0801.05014 [48] Pott, A., A survey on relative difference sets, (), 195-232 · Zbl 0847.05018 [49] Schmidt, B., On (p^a,pb,pa,pab)-relative difference sets, J. algebraic combin., 6, 279-297, (1997) [50] B. Schmidt, Nonexistence results on Chen and Davis-Jedwab difference sets, J. Algebra · Zbl 0910.05014 [51] Seberry, J.; Yamada, M., Hadamard matrices, sequences, and block designs, (), 431-560 · Zbl 0776.05028 [52] Skinner, G.K., X-ray imaging with coded masks, Scientific American, 259, 66-71, (Aug. 1988) [53] Smith, K.W., Non-abelian Hadamard difference sets, J. combin. theory ser. A, 70, 144-156, (1995) · Zbl 0818.05015 [54] Spence, E., A family of difference sets, J. combin. theory ser. A, 22, 103-106, (1977) · Zbl 0357.05019 [55] Turyn, R.J., Character sums and difference sets, Pacific J. math., 15, 319-346, (1965) · Zbl 0135.05403 [56] Turyn, R.J., A special class of williamson matrices and difference sets, J. combin. theory ser. A, 36, 111-115, (1984) · Zbl 0523.05016 [57] Wilson, R.M.; Xiang, Q., Constructions of Hadamard difference sets, J. combin. theory ser. A, 77, 148-160, (1997) · Zbl 0868.05010 [58] Xia, M.-Y., Some infinite classes of special williamson matrices and difference sets, J. combin. theory ser. A, 61, 230-242, (1992) · Zbl 0772.05022 [59] Xiang, Q.; Chen, Y.Q., On Xia’s construction of Hadamard difference sets, Finite fields appl., 2, 87-95, (1996) · Zbl 0844.05020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.