The sizes $k$ of the complete $k$-caps in PG$(n,q)$, for small $q$ and $3\le n\le 5$

Abstract

It is known that there exists a one-to-one correspondence between the classes of equivalent $[n,n-k,4]$-codes over $\mathrm{G}\mathrm{F}(q)$ and the classes of projectively equivalent complete $n$-caps in $\mathrm{P}\mathrm{G}(k-1,q)$ (see [{20}], [{40}]). Hence all results on caps can be translated in terms of such codes. This fact stimulated many researches on the fundamental problem of determining the spectrum of the values of $k$ for which there exist complete $k$-caps in $\mathrm{P}\mathrm{G}(n,q)$. This paper reports the result of a computer search for the spectrum of $k$’s that occur as a size of a complete $k$-cap in some finite projective spaces. The full catalog of such sizes $k$ is given in the following projective spaces: $\mathrm{P}\mathrm{G}(3,q)$, for $q\le 5$, $\mathrm{P}\mathrm{G}(4,2)$, $\mathrm{P}\mathrm{G}(4,3)$, $\mathrm{P}\mathrm{G}(5,2)$. Concrete examples of such caps are presented for each possible $k$.${}^{\ast }$