On small line sets with few odd-points

Abstract

In this paper, we study small sets of lines in $${{\mathrm{PG}}}(n,q)$$ PG ( n , q ) and $${{\mathrm{AG}}}(n,q),\,q$$ AG ( n , q ) , q odd, that have a small number of odd-points. We fix a small glitch in the proof of an earlier bound in the affine case, we extend the theorem to the projective case, and we attempt to classify all the sets where equality is reached. For the projective case, we obtain a full classification. For the affine case, we obtain a full classification minus one open case where there is only a characterization.