On Srivastava's problems and the properties of Hadamard matrices (Q1356560)

Let \(K\) be a \((0,1)\) matrix of size \((n,m)\), \(w_m\) the set of integers \(1,2,\dots,m\) and \(W_m\) the class of all subsets of \(w_m\). Let \(A=A(k)\) be a \((0,1)\) matrix of size \((n,2^m)\) whose columns are labeled by elements of \(W_m\). The column labeled by \(\{i\}\) equals the \(i\)th column \(K_i\) of \(K\) \((i=1,\dots,m)\) and the column labeled by a subset \(I\) equals the sum of \(K_i\), where \(i\) runs over the elements of \(I\). If \(I\) is empty, then the corresponding column of \(A\) is zero. \(A^*= A^*(K)\) is a real matrix obtained from \(A\) by replacing 0 and 1 by 1 and \(-1\), respectively. \(P_t\) is a property for a real matrix \(B\) which says every set of \(t\)-columns of \(B\) is linearly independent. Around 1980, J. Srivastava asked the following question. What is the minimum value of \(n\), denoted by \(N(t,m)\), for which there exists a \((0,1)\) matrix \(K\) of size \((n,m)\) such that the corresponding matrix \(A^*\) has the property \(P_t\)? Srivastava then obtained such a \(K\). In this paper, the authors show that \(N(t,m)= m+1\) for \(t=2\) and 3, \(N(t,m)= 2^m-1\) for \(t\) between \(2^{m-1}\) and \(2^m-1\), and \(N(t,m)=2^m\) for \(t=2^m\). In particular, they solve the problem for \(t=2\) and 3.