On Srivastava's problems and the properties of Hadamard matrices

Abstract

The Hadamard matrix H_2^m of order 2^m 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 Srivastava's problems for positive integers t and m is equivalent to finding a submatrix $\text{A}{}^{\mathrm{\ast }}$ consisting of N(t, m) rows of the H_2^m such that the number N(t, m) of rows in minimal and any t columns of $\text{A}{}^{\mathrm{\ast }}$ are linearly independent. For 2 ⩽ t ⩽ 3 and 2^m−1 ⩽ t ⩽ 2^m, the minimum number N(t, m) of rows of $\text{A}{}^{\mathrm{\ast }}$ is given and the method for constructing $\text{A}{}^{\mathrm{\ast }}$ is presented. For 4 ⩽ t < 2^m−1 we point out the upper bound on N(t, m) and conjecture that this upper bound is the minimum number of rows of $\text{A}{}^{\mathrm{\ast }}$.