A characterization of flat spaces in a finite geometry and the uniqueness of the hamming and the MacDonald codes

Abstract

Let PG(k−1, q) be the projective geometry of dimension k−1 over the finite field GF(q) where q is a prime power. The flat spaces of PG(k−1, q) may be characterized by the following Theorem. If F is a set of points in PG( k−1, q) which has a non-empty intersection with every v-flat, then the number of points in F is greater than or equal to ( q k−v−1 )/( q−1). Equality holds if, and only if, F is a ( k−v, −1)- flat. It may be shown from this theorem that the Hamming codes which maximize n for a given redundancy r, q=2, and minimum distance d=4, are unique. An extension of the theorem shows that the MacDonald codes with d=q k−1 − q u ( u=0, 1, …, k−2) are unique.