On the inclusion matrix W23(v)

For a v-set X, W23(v) is a (v2)×(v3) inclusion matrix where rows and columns are indexed by pairs and triples of X, respectively, and for row T and column K, W23(v)(T,K)=1 if T⊆K and zero otherwise. In this paper, we classify the basis elements of the nullZ(W23(v)), derived from the Gaussian elimination on W23(v) (called standard basis), into five classes. Then, we present a new algorithm to construct a (2,3,v)-halving for a feasible v, i.e. a nowhere zero T(2,3,v)-trade.