On Srivastavaʹs problems and the properties of Hadamard matrices

The Hadamard matrix H2m of order 2m can be obtained by m−1 times of Kronercker products from the Hadamard matrix H2 of order 2. In this paper we first point out that the Srivastavaʹs problems for positive integers t and m is equivalent to finding a submatrix A∗ consisting of N(t, m) rows of the H2m such that the number N(t, m) of rows in minimal and any t columns of A∗ are linearly independent. For 2 ⩽ t ⩽ 3 and 2m−1 ⩽ t ⩽ 2m, the minimum number N(t, m) of rows of A∗ is given and the method for constructing A∗ is presented. For 4 ⩽ t < 2m−1 we point out the upper bound on N(t, m) and conjecture that this upper bound is the minimum number of rows of A∗.