The Cardinality of Sets of k -Independent Vectors over Finite Fields

A set of vectors is k-independent if all its subsets with no more than k elements are linearly independent. We obtain a result concerning the maximal possible cardinality Ind _q (n, k) of a k-independent set of vectors in the n-dimensional vector space F _q ⁿ over the finite field F _q of order q. Namely, we give a necessary and sufficient condition for Ind _q (n, k) = n + 1. We conclude with some pertinent remarks re applications of our results to codes, graphs and hypercubes.