An Analysis of the Lines in the Three Dimensional Affine Space Over $F_3$

Abstract

Let $A(n,3)$ denote the $n$-dimensional affine space over the finite field of order three. In this paper, we use basic combinatorial principles to discuss some old and new results about the lines in $A(3,3)$. For $S\subset A(3,3)$, let $||S|{|}_3$ and $||S|{|}_{3,k}$ respectively denote the number of lines and the number of $k$-lines of $A(3,3)$ contained entirely in $S$. For each $t$, we compute ${\alpha }_3(t)=min\{||S|{|}_3:|S|=t\}$ and ${\mathrm{\Omega }}_3(t)=max\{||S|{|}_3:|S|=t\}$. We also give results about ${\alpha }_{3,k}(t)=min\{||S|{|}_{n,k}:|S|=t\}$ and ${\omega }_{3,k}(t)=max\{||S|{|}_{n,k}:|S|=t\}$ and results about $1$-lines and $n$-lines in $A(n,3)$.