On a Bound Useful in the Theory of Factorial Designs and Error Correcting Codes

Abstract

Consider a finite projective space $PG(r - 1, s)$ of $r - 1$ dimensions, $r \geqq 3$, based on the Galois field $GF_s$, where $s = p^h, p$ being a prime. A set of distinct points in $PG(r - 1, s)$ is said to be a non-collinear set, if no three are collinear. The maximum number of points in such a non-collinear set is denoted by $m_3(r, s)$. It is the object of this paper to find a new upper bound for $m_3(r, s)$. This bound is of importance in the theory of factorial designs and error correcting codes. The exact value of $m_3(r, s)$ is known when either $r \leqq 4$ or when $s = 2$. When $r \geqq 5, s > 3$, the best values for the upper bound on $m_3(r, s)$ are due to Tallini [10] and Barlotti [1]. Our bound improves these when $s = 3$ or when $s$ is even.