Dimensions of hyperplane spaces over finite fields

Abstract

In this paper we shall determine the dimensions of certain function spaces associated with the hyperplanes in a vector space over a finite field. (The results obtained are summarized in Table l.) This investigation was motivated by the desire to obtain bounds on the size of certain hyperplane coverings as discussed in [7], and the principal result here will be a proof of the dimension formula asserted at the end of that paper. Although some further applications to hyperplane coverings and convexity over finite fields will be given in a later paper, the results obtained here seem to be of independent interest and the exposition is self-contained. To more precisely describe the problem to be solved here, let V be a d dimensional vector space over a finite field F with q = p e elements (p=charF) . By a hyperplane in V we mean any translate (that is, coset) of any d 1 dimensional subspace of V. The hyperplanes in V fall naturally into two classes: those which contain the zero vector (and hence are linear subspaces) and those which do not. For brevity, a hyperplane not containing 0 will be called a nonzero hyperplane. Now associate with each hyperplane in V its characteristic function. As this function takes only the values 0 and 1, it may be regarded as a function from V into any arbitrary field K. Of course, with the usual pointwise operations, the set K v of all functions from V into the field K is a K-vector space. Thus one may define, for any coefficient field K, the following hyperplane spaces over V: