An Algorithm for Enumerating Trades in Designs, with an Application to Defining Sets

Abstract

An algorithm is presented which, when given the non-isomorphic designs with given parameters, generates all the trades in each of the designs. The lists of trades generated by the algorithm were used to find the sizes, previously unknown, of smallest defining sets of the $21$ non-isomorphic $2$-(10, 5, 4) designs. Consideration of trades in a design to isomorphic and to non-isomorphic designs led to two variations on the concept of defining sets. The lists of trades were then used to find the sizes of these smallest member and class defining sets, for five parameter sets.