The sizes \(k\) of the complete \(k\)-caps in PG\((n, q)\), for small \(q\) and \(3 \leq n\leq5\)

It is known that there exists a one-to-one correspondence between the classes of equivalent \([n, n-k, 4]\)-codes over \(\mathrm{GF}(q)\) and the classes of projectively equivalent complete \(n\)-caps in \(\mathrm{PG}(k-1, q)\) (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 \(\mathrm{PG}(n, q)\). 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: \(\mathrm{PG}(3, q)\), for \(q \leq 5\), \(\mathrm{PG}(4, 2)\), \(\mathrm{PG}(4, 3)\), \(\mathrm{PG}(5, 2)\). Concrete examples of such caps are presented for each possible \(k\).\(^*\)