A Geometric Construction of (Ƙ, r)-cap in PG(3, q) for q prime, 2 ≤q ≤997

Abstract

The objective of this paper is to designing computer programs to find points, planes, and lines, as well as to find mr(3, q) for the projective space PG(3,q), when q is prime, and a geometric method for constructing of a Complete (Ƙ, r)-cap; when q is prime, 2 ≤ q ≤ 997, r = 2, and found the maximum complete (Ƙ, r)-cap, we took PG(3,17) as an example, where we found a Complete (Ƙ, r)-cap; 2 ≤ r ≤ 18. We found that the maximum complete (Ƙ, r)-cap which is called an ovaloid, exists in PG(3,17) when Ƙ = 67. Moreover the complete (Ƙ,3)-caps, (Ƙ,4)-caps, (Ƙ,5)-caps, (Ƙ,6)-caps, (Ƙ,7)-caps, (Ƙ,8)-caps, (Ƙ,9)-caps, (Ƙ,10)-caps, (Ƙ,11)-caps, (Ƙ,12)-caps, (Ƙ,13)-caps, (Ƙ,14)-caps, (Ƙ,15)-caps, (Ƙ,16)-caps, (Ƙ,17)-cap and (Ƙ,18)-caps in PG(3,17).