An Analysis of the Lines in the Three Dimensional Affine Space Over \(F_3\)

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 \(\Omega_3(t) = \max\{||S||_3 : |S| = t\}\). 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 \({A}(n,3)\).