On the linear independence of sets of 2^q columns of certain (1, −1) matrices with a group structure, and its connection with finite geometries

Abstract

Consider a set of m symbols (indeterminates) F₁, ..., F_m, and let G be the group of order 2^m generated by multiplying these symbols two, or three, or more at a time, where the multiplication is assumed commutative, and where F ²_j =μ (the identity element of G) for all j. The elements of G can be written, in order, as {μ; F₁, ..., F_m; F₁F₂,F₁F₃,...,F_m−1F_m;F₁F₂F₃,...;F₁F₂ ... F_m}. Consider a matrix A(N × 2^m) over the real field whose columns correspond in order to the elements of the group G. The elements of A are 1 and (−1), and are obtained as follows. The elements of A in the column corresponding to μ are all equal to 1. The next m columns of A, filled in arbitrarily, constitute an (N × m) submatrix, say A*. Finally, for all ℓ (1 ≤ ℓ ≤ m), and all i₁,...,i_ℓ (with 1 ≤ i₁<i₂<...<i_ℓ ≤ m), the column of A corresponding to F_{i 1} F_{i 2} ...F_{i ℓ} is obtained by taking the Schur product of the columns of A (or A*) corresponding to F_{i 1},F_{i 2},...,F_{i ℓ}. The matrix A (over the real field) is said to have the property P_t if and only if every set of t columns of A is linearly independent. In this paper, for all positive integers q, we obtain necessary conditions on A* such that every (N × 2^q) submatrix A** in A has rank 2^q. A non-statistical introduction together with an illustrative example is provided.