A recursive construction of 1-rotational Steiner 2-designs
A Steiner 2-design S(2,k,v) is a collection \({\mathcal B}\) of k-subsets (blocks) of \(Z_ v\) (the set of residues modulo v) such that every pair of elements of \(Z_ v\) is contained in exactly one block. In particular, a Steiner 2-design RS(2,k,v) is said to be 1-rotational if its block are k-subsets of \(Z_{v-1}\cup \{\infty \}\) such that for any block B and \(y\in Z_{v-1}\) the sets \(B+y=\{b+y(mod(v-1)):b\in B\}\) are also blocks. The fixed point \(\infty\) operates with the elements of \(Z_{v-1}\) as: \(y+\infty =\infty +y=\infty\) for every \(y\in Z_{v-1}\). For example, let \(B_ 1=\{0,4,8\}\) and \(B_ 2=\{0,1,3\}\); then \({\mathcal B}=\{B_ 1+y:y\in Z_ 8\}\cup \{B_ 2+y:y\in Z_ 8\}\) is an RS(2,3,9) (here 8 represents the fixed point \(\infty).\) The paper presents a recursive construction for a wide class of 1- rotational Steiner 2-designs, using only elementary methods.
scientific article; zbMATH DE number 3912396