4-blocked Hadamard 3-designs
A \(t\)-\((\nu,k, \lambda)\) design \(D\) is a pair \((V,B)\), where \(V\) is a set with \(\nu\) elements and \(B\) is a collection of \(k\)-subsets of \(V\) (called blocks), such that any subset of \(t\) elements of \(V\) appears in exactly \(\lambda\) blocks. A blocking \(c\)-set in \(D\) is a set \(C\) with \(c\) elements in it such that any block has in it a point of \(C\) and a point off \(C\). An Hadamard 3-design \(D_3\) of order \(n\) is a 3-\((4n, 2n,n-1)\) design whose existence is linked with the existence of an Hadamard matrix of size \(4n\). In this paper are investigated 4-blocked Hadamard 3-designs, i.e. Hadamard 3-designs that contain blocking 4-sets. The main result is as follows: Theorem. An Hadamard 3-design of odd order \(n\) is 4-blocked if and only if it contains a special \((n-1)\)-tuple. Moreover, if an Hadamard 3-design of even order \(n\) is 4-blocked, then it contains a special \(n\)-tuple. According to the above theorem the existence of blocking 4-sets is connected to the existence of special \(n\)-tuples or \((n-1)\)-tuples.
scientific article; zbMATH DE number 1110070
scientific article