Maximal arcs in designs

Abstract

An (α,n)-arc in a 2-design is a set ofn points of the design such that any block intersects it in at most α points. For such an arc,n is bounded by 1+(r(α−1)/λ), with equality if and only if every block meets the arc in either 0 or α points. An (α,n) arc with equality in above is said to be maximal. A maximal block arc can be dually defined. This generalizes the notion of an oval (α=2) in a symmetric design due to Asmus and van Lint. The aim of this paper is to study the infinite family of possibly extendable symmetric designs other than the Hadamard design family and their related designs using maximal arcs. It is shown that the extendability corresponds to the existence of a proper family of maximal arcs. A natural duality between point and block arcs is established, which among other things implies a result of Cameron and van Lint that extendability of a given design in this family is equivalent to extendability of its dual. Similar results are proved for other related designs.