SBIBD(4k\({}^ 2,2k^ 2+k,k^ 2+k)\) and Hadamard matrices of order \(4k^ 2\) with maximal excess are equivalent
The sum of the elements of an Hadamard matrix \({\mathcal H}\) is denoted by o(\({\mathcal H})\) and called excess of \({\mathcal H}\). \(o(n)=\max o({\mathcal H})\) for all Hadamard matrices of order n. It is well-known that o(n)\(\leq n\sqrt{n}\). In this paper the following statement is proved. Theorem. There is an Hadamard matrix of order \(4s^ 2\) and maximal excess \(8s^ 2\) if and only if there is an \(SBIBD(4s^ 2,2s^ 2+s,s^ 2+s).\) It is shown that \(SBIBD(4s^ 2,2s^ 2+s,s^ 2+s)\) is equivalent to existence of a regular Hadamard matrix of order \(4s^ 2\) whith maximal excess.
scientific article; zbMATH DE number 4172992
scientific article