Algebraic completions of SBIBD

We study the minimal set of vectors required to make the (0,1) incidence matrix of an SBIBD(v,k,r). We devise algorithms to generate all possible vectors which could complete SBIBD and to find minimal sets of such vectors. The 2-(7,3,l) and 2-( 16,6,2) designs were studied to test our algorithms on the work of Greenhill and Street. We show the minimal defining set of the 5i5/BD(31,15, 7) constructed using the Paley difference set has 13 blocks. We give a minimal defining set with 19 blocks for the SBIBD (31, 15, 7) constructed using the Hall difference set. We show the smallest minimal defining set of teh Hall SBIBD(31,15,7) has between 16 and 20 elements. We investigate the influence of the removal of vectors from minimal sets and discuss the relationship with secret sharing schemes.