258 V Hadamard’s Determinant Problem
(ii) If there exists av×vmatrixAof integers such that
AAt=(k−λ)I+λJ,JA=kJ,then every entry ofAis either 0 or 1, and thusAis the incidence matrix of a square
2-design.
For introductions to the classification theorem for finite simple groups, see
Aschbacher [2] and Gorenstein [15]. Detailed information about the finite simple
groups is given in Conwayet al.[10]. There is a remarkable connection between
the largest sporadic simple group, nicknamed the ‘Monster’, and modular forms; see
Ray [32].
Good introductions to coding theory are given by van Lint [24] and Pless [30].
MacWilliams and Sloane [26] is more comprehensive, but less up-to-date. Assmus and
Mattson [3] is a useful survey article. Connections between codes, designs and graphs
are treated in Cameron and van Lint [8]. The historical account in Thompson [36]
recaptures the excitement of scientific discovery.
9 SelectedReferences
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[2] M. Aschbacher, The classification of the finite simple groups,Math. Intelligencer 3
(1980/81), 59–65.
[3] E.F. Assmus Jr. and H.F. Mattson Jr., Coding and combinatorics,SIAM Rev. 16 (1974),
349–388.
[4] M. Barnabei, A. Brini and G.-C. Rota, On the exterior calculus of invariant theory,
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[5] T. Beth, D. Jungnickel and H. Lenz,Design theory, 2nd ed., 2 vols., Cambridge University
Press, 1999.
[6] N. Bourbaki,Algebra I, Chapters 1–3, Hermann, Paris, 1974. [French original, 1948]
[7] A.E. Brouwer, Block designs,Handbook of combinatorics(ed. R.L. Graham, M. Gr ̈otschel
and L. L ́ovasz), Vol. I, pp. 693–745, Elsevier, Amsterdam, 1995.
[8] P.J. Cameron and J.H. van Lint,Designs, graphs, codes and their links, Cambridge
University Press, 1991.
[9] P.M. Cohn,Algebra, 2nd ed., Vol. 3, Wiley, Chichester, 1991.
[10] J.H. Conwayet al.,Atlas of finite groups, Clarendon Press, Oxford, 1985.
[11] R. Craigen and W.D. Wallis, Hadamard matrices: 1893–1993,Congr. Numer. 97 (1993),
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[12] F.R. Gantmacher,The theory of matrices, English transl. by K. Hirsch, 2 vols., Chelsea,
New York, 1960.
[13] I.M. Gelfand and V.S. Retakh, A theory of noncommutative determinants and character-
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[14] G.H. Golub and C.F. van Loan,Matrix computations, 3rd ed., Johns Hopkins University
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[15] D. Gorenstein, Classifying the finite simple groups,Bull. Amer. Math. Soc.(N.S.) 14
(1986), 1–98.
[16] J. Hadamard, R ́esolution d’une question relative aux determinants,Selecta, pp. 136–142,
Gauthier-Villars, Paris, 1935 andOeuvres, Tome I, pp. 239–245, CNRS, Paris, 1968.
[17] M. Hall,Combinatorial theory, 2nd ed., Wiley, New York, 1986.