The Number of Linearly Independent Binary Vectors with Applications to the Construction of Hypercubes and Orthogonal Arrays, Pseudo ( t, m, s)-Nets and Linear Codes

We study formulae to count the number of binary vectors of length n that are linearly independent k at a time where n and k are given positive integers with 1kn. Applications are given to the design of hypercubes and orthogonal arrays, pseudo (t, m, s)-nets and linear codes.