A series of Hadamard designs with large automorphism groups (Q1841842)

A symmetric design with parameters \((v,k,\lambda)\) is a special case of the balanced incomplete block design \((v,b,r,k,\lambda)\) in which \(v= b\), and \(r=k\). The parameters of a symmetric \((v,k,\lambda)\)-design satisfy the condition \(\lambda(v- 1)= k(k-1)\). A Hadamard design is a symmetric design with parameters \((4n-1, 2n-1, n- 1)\) and is closely related to a Hadamard matrix of order \(4n\) with elements \(\{1,-1\}\). Let \(q\) be an arbitrary odd prime power. The authors present a general construction for a series of Hadamard designs with parameters \((2q^2+ 1, q^2, {q^2-1\over 2})\) which admit an action of the elementary abelian graph of order \(q^2\) with three orbits on points and blocks of lengths \(1\), \(q^2\), \(q^2\). The following is the main result. Result: (i) For an odd prime \(p\) let \(G\times G\) be the direct product of the Frobenius group \(G\) of order \({p(p- 1)\over 2}\) with itself. Then, this group is an automorphism group of the symmetric \((2p+ 1, p^2, {p^2- 1\over 2})\)-design (as defined in the paper). (ii) For an odd prime power \(q\), let \(E\) be the elementary abelian group of order \(q^2\). Then, this group is an automorphism group of the symmetric \((2q^2+ 1, q^2, {q^2- 1\over 2})\)-design (as defined in the paper).