The Spectrum of Values k for Complete $k$-Arcs in PG$(2,q)$ for $q\le 23$

Abstract

Arcs and linear maximum distance separable $(M.D.S.)$ codes are equivalent objects~$[25]$. Hence, all results on arcs can be expressed in terms of linear M.D.S. codes and conversely. The list of all complete $k$-arcs in $\mathrm{P}\mathrm{G}(2,q)$ has been previously determined for $q\le 16$. In this paper, (i) all values of $k$ for which there exists a complete $k$-arc in $\mathrm{P}\mathrm{G}(2,q)$, with $17\le q\le 23$, are determined; (ii) a complete $k$-arc for each such possible $k$ is exhibited.