The maximum number of trades of volume four in a symmetric design

Abstract

Khosrovshahi, Majumdar and Widel, building on an earlier result by Hwang, have characterized the structure of trades of volume four contained in a symmetric design. This characterization is used here to find a complete set of non-isomorphic smallest defining sets of one of the (16,6,2) designs. It is also shown that a symmetric design over v elements contains at most v(v - 1)(v - 3)/24 distinct trades of volume four and that this bound is attained precisely when the blocks of the design are the hyperplanes of the protective space PG(d,2).