Pair-Splitting Sets in $AG ( m,q )$

Abstract

This paper is concerned with pair-splitting sets in $AG_k ( m,q )$, the design obtained from the points and k-flats in $AG (m,q)$. A pair-splitting set is a set of parallel classes $\{ R_1 ,R_2 , \cdots ,R_s \}$ such that there is no pair of distinct points $a,b$ such that $a,b$ are contained in a common k-flat of each of the s parallel classes. It is easy to prove that a lower bound on s is $\lceil m/( m - k ) \rceil $. The main result of this paper is to prove that this lower bound is always achievable for any choice of $m,q$, and k, where $0\leqq k\leqq m - 1$. The concept of pair-splitting sets arises naturally out of the problem of finding roots in $GF( q^m )$ of a polynomial over $GF ( q^m )$. The connection between the two concepts is briefly discussed.