New good large $(n, r)$-arcs in PG(2,29) and PG(2,31)

Abstract

An $(n, r)$-arc is a set of $n$ points of a projective plane such that some $r$, but no $r+1$ of them, are collinear. The maximum size of an $(n, r)$-arc in PG$(2,q)$ is denoted by $m_r(2,q)$. In this article a $(477, 18)$-arc, a $(596,22)$-arc, a $(697,25)$-arc in PG(2,29) and a $(598, 21)$-arc, a $(664, 23)$-arc, a $(699, 24)$-arc, a $(769, 26)$-arc, a $(838,28)$-arc in PG(2,31) are presented. The constructed arcs improve the respective lower bounds on $m_r(2,29)$ and $m_r(2,31)$ in [6]. As a consequence there exist eight new three-dimensional linear codes over the respective finite fields.

Received: 12 December 2022 | Accepted: 30 April 2023