Abstract
The incidence matrix W-t k is defined as follow: Let V be a finite set, with v elements. Given non-negative integers t, k, W-t k is the ((t) (v)) by ((v)(k)) matrix of 0's and 1's, the rows of which are indexed by the t-element subsets T of V, the columns are indexed by the k-element subsets K of V, and where the entry W-t k(T, K) is 1 if T subset of K and is 0 otherwise.
R.M. Wilson proved that for t <= min(k, v - k), the rank of Wt k modulo a prime p is Sigma(t)(i=0) ((v)(i)) - ((v)(i-1)) where p does not divide the binomial coefficient ((k-i)(t- i)).
In this paper, we begin by giving an analytic expression of the rank of the matrix Wt k when t = t(0) + t(1p) + t(2)p(2), with t(0), t(1), t(2) is an element of [0, p - 1] and we characterize values of t and k such that dim Ker(W-t(t k)) is an element of {0, 1}. Next, using this result we generalize a result in the (<= 6)-reconstruction of digraphs due to G. Lopez.