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Note on Hadamard matrices of type Q. (English) Zbl 0537.05012
Let u be an odd integer, \(n=4u\) and \(H=(h_{ij})\) a Hadamard matrix of order n. Let P be the set of 2n points \(1,2,...,n,1^*,2^*,...,n^*\). Define an n-subset \(\alpha_ i\) of P by \(j\in a_ i\) if \(h_{ij}=+1\), \(j^*\in a_ i\) if \(h_{ij}=-1\), 1\(\leq i\), \(j\leq n\). \(\alpha^*_ i=P-\alpha_ i\). \(\alpha_ i\) and \(\alpha^*_ i\) are blocks and B is the set of 2n blocks \(\alpha_ 1,\alpha_ 2,...,\alpha^*_ 1,\alpha^*_ 2,...,\alpha^*_ n\). \(M(H)=(P,B)\) is the matrix design of H. Let G be the set of all permutations \(\sigma\) on P which induce a permutation on B, then G is the automorphism group M(H) and isomorphic to the automorphism group of H. The paper defines R to be the permutation group on P generated by three given permutations. R has order 8u, a cyclic normal subgroup of order u, a Sylow 2-subgroup which is isomorphic to the quaternion group and is regular transitive on P. H is called a Hadamard matrix of type Q if R is regular transitive on B. The paper proves the following pretty results: 1) The Hadamard matrix of quadratic residue type of order \(n=4u=q+1\), \(q\equiv 3(mod 8)\), q a prime power is a Hadamard matrix of type Q; 2) a Hadamard matrix of Williamson type using symmetric circulants is a Hadamard matrix of type Q. In particular, its automorphism group contains a regular transitive subgroup; 3) a Hadamard matrix of Paley type of order \(n=4u=2(q+1)\), \(q\equiv 1(mod 4)\), q a prime power is a Hadamard matrix of type Q.
Reviewer: Jennifer Seberry

05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures