On the original Steiner systems
An original Steiner system (OSS) St(k,v), \(k\leq v\) is a pair (X,\({\mathcal B})\), where X is a finite set and \({\mathcal B}\) is a family of subsets - called blocks - of X, satisfying the following conditions: (i) \(| X| =v\); (ii) \({\mathcal B}=\cup^{k}_{n=3}{\mathcal B}(n)\) and \(| B_ i| =n\) for every \(B_ i\in {\mathcal B}(n);\) (iii) every pairset \(\{\) x,y\(\}\) \(\subset X\) is contained in exactly one block of \({\mathcal B}(3)\); (iv) every i-subset of X, \(3\leq i\leq k-1,\) which does not contain any block of \(\cup^{i}_{j=3}{\mathcal B}(j)\) is contained in exactly one block of \({\mathcal B}(i+1)\); no block of \({\mathcal B}(i+1)\) contains as subsets blocks of \(\cup^{i}_{j=3}{\mathcal B}(j)\). As the author points out Steiner knew that \(| {\mathcal B}(i)| =\prod^{i-2}_{j=0}(v+1-2^ j)/i!,\quad 3\leq i\leq k,\) and accordingly all these numbers must be integers. The author proves that for every \(k\geq 3\) an OSS \(St(k,2^{k-1}-1)\) exists. It is obtained by choosing \(X=GF(2^{k-1})\setminus \{0\}.\) \(H\subset X\), \(| H| \geq 2\) is said to be independent if the sum of elements of any nonempty subset of H is different from zero. H is closed by adjoining to it the sum of its elements. The sets of the closure of H form the family \({\mathcal B}\) of blocks of the \(St(k,2^{k-1}- 1)\). Moreover since \(| {\mathcal B}(k+1)| =0\) the design may be denoted \(St(k+1,2^{k-1}-1)\) with \({\mathcal B}(k+1)=\emptyset\).
scientific article; zbMATH DE number 3884179