A new proof of a classical theorem in design theory (Q5930031)

Given integers \(0\leq t\leq k\leq v\), the inclusion matrix \(W_{tk}\) is the \(0, 1\) matrix whose rows are indexed by the \(t\)-subsets of a \(v\)-set \(X\), whose columns are indexed by the \(k\)-subsets of a \(X\) and whose \(T\), \(K\) entry is \(1\) if and only if \(T\subseteq K\). An integral solution \(x\) of the equation \(W_{tk} x=\lambda e\), where \(\lambda\) is a positive integer and \(e\) is the column vector of \(1\)'s, is called a signed \(t\)-\((v,k,\lambda)\) design. The authors give a new proof to the following theorem:NEWLINENEWLINENEWLINETheorem 1. A signed \(t\)-\((v,k,\lambda)\) design exists if and only if NEWLINE\[NEWLINE\lambda{v-i\choose t-i}\equiv 0\text{ mod}{k-i\choose t-i},\qquad\text{for}\quad i= 0,\dots,t.NEWLINE\]