# Hadamard matrices/1. A Hadamard matrix is an n n matrix H with entries ±1 which satisfies HH = ni. The name derives from a theorem of Hadamard:

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3 it is a Hadamard matrix. For example, S(2) = The Sylvester matrices have many other descriptions. For example, if we index the rows and columns by all k-tuples over the binary field GF(2), we can take the entry in row a = (a 1,...,a k ) and column b = (b 1,...,b k ) to be ( 1) a b, where a b = a i b i is the usual dot product of vectors. We can regard the index (a 1,...,a k ) as being the base 2 representation of an integer a i 2 k i in the range [0,2 k 1]. Alternatively, S(k) is the character table of the elementary abelian group of order 2 k. 2.2 Paley matrices Let q be a prime power congruent to 3 mod 4. Recall that in the field GF(q), half the non-zero elements are quadratic residues or squares, and half are quadratic non-residues or non-squares; and in particular, +1 is a square and 1 is a nonsquare. The quadratic character of GF(q) is the function χ given by { 0 if x = 0; χ(x) = +1 if x is a quadratic residue; -1 if x is a quadratic non-residue. Now let A be the matrix whose rows and columns are indexed by elements of GF(q), and having (x,y) entry a xy = χ(y x). The matrix A is skew-symmetric, with zero diagonal and ±1 elsewhere, and satisfies the equation A 2 = J qi. The matrix A is the adjacency matrix of the Paley tournament. Now if we replace the diagonal zeros by 1s and border A with a row and column of +1s, we obtain a Hadamard matrix of order q+1 called a Paley matrix. 2.3 Order 36 The smallest order not settled by the above constructions is 36. Here are two completely different methods for constructing Hadamard matrices of order 36. Hadamard matrices/3

4 Construction from Latin squares Let L = (l i j ) be a Latin square of order 6: that is, a 6 6 array with entries 1,...,6 such that each entry occurs exactly once in each row or column of the array. Now let H be a matrix with rows and columns indexed by the 36 cells of the array: its entry in the position corresponding to a pair (c,c ) of distinct cells is defined to be +1 if c and c lie in the same row, or in the same column, or have the same entry; all other entries (including the diagonal ones) are 1. Then H is a Hadamard matrix. Steiner triple systems Let S be a Steiner triple system of order 15: that is, S is a set of triples or 3-element subsets of {1,...,15} such that any two distinct elements of this set are contained in a unique triple. There are 35 triples, and two distinct triples have at most one point in common. Now let A be a matrix with rows and columns indexed by the triples, with entry in position (t,t ) being 1 if t and t meet in a single point; all other entries (including the diagonal ones)are +1. Now let H be obtained by bordering A by a row and column of +1s. Then H is a Hadamard matrix. 3 Equivalence of Hadamard matrices The Sylvester and Paley matrices of orders 4 and 8 are equivalent indeed there is essentially a unique matrix of each of these orders. For all larger orders for which both types exist (that is, n = p + 1, where p is a Mersenne prime), they are not equivalent. We proceed to make the sense of equivalence of Hadamard matrices precise. There are several operations on Hadamard matrices which preserve the Hadamard property: (a) permuting rows, and changing the sign of some rows; (b) permuting columns, and changing the sign of some columns; (c) transposition. We call two Hadamard matrices H 1 and H 2 equivalent if one can be obtained from the other by operations of types (a) and (b); that is, if H 2 = P 1 H 1 Q, where P and Q are monomial matrices (having just one non-zero element in each row or column) with non-zero entries ±1. Accordingly, the automorphism group of a Hadamard matrix H is the group consisting of all pairs (P, Q) of monomial matrices with non-zero entries ±1 Hadamard matrices/4

6 This is the Paley design. Yet another design can be obtained as follows. Let H be a Hadamard matrix of order 4a. The points of the design are the columns of H; for each pair of rows of H, there are two blocks of size 2a, the set of columns where the entries in the rows agree, and the set where they disagree. This is a 3-(4a,2a,2a(a 1)) design. Equivalent matrices give the same design. Remarkably it turns out that the design is a 4-design if and only if a = 3, in which case it is even a 5-design (specifically the 5-(12, 6, 1) Steiner system, whose automorphism group is the Mathieu group M 12 which we met above). 5 Symmetric matrices with constant row sum If a Hadamard matrix H is symmetric with constant row sum, then its order is a square, say 4m 2, and the row sum is either 2m or 2m. If we replace the entries 1 in the matrix by 0, we obtain the incidence matrix of a square 2-(4m 2,2m 2 ± m,m 2 ± m) design. Any Sylvester matrix of square order is equivalent to a symmetric matrix with constant row sum, and thus gives rise to such designs; these can be constructed using quadratic forms on a vector space over GF(2). The Hadamard matrices of order 36 constructed above from Latin squares are also of this form. References [1] M. Hall, Jr., Note on the Mathieu group M 12, Arch. Math. 13 (1962), [2] Institute for Studies in Theoretical Physics and Mathematics, IPM, Iran; [3] H. Kharaghani and B. Tayfeh-Rezaie, A Hadamard matrix of order 428, J. Combinatorial Designs 13 (2005), [4] J. Seberry and M. Yamada, Hadamard matrices, sequences and block designs, pp in Contemporary Design Theory: A Collection of Surveys (ed. J. H. Dinitz and D. R. Stinson), Wiley, New York, [5] Neil Sloane, A library of Hadamard matrices; njas/hadamard/ Hadamard matrices/6

7 Peter J. Cameron June 8, 2006 Hadamard matrices/7

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