On complete caps different from ovaloids of the space \(PG(3,q)\), \(q\) odd (Q1389826)

In this article, complete caps in the 3-dimensional projective space \(PG(3,q)\) over a finite field \(GF(q)\) of odd characteristic are constructed. A \(k\)-cap in \(PG(3,q)\) is a set of \(k\) points, no three of which are collinear. A \(k\)-cap is called complete when it is not contained in a \((k+1)\)-cap of \(PG(3,q)\). For \(q\) odd, the maximal size of a \(k\)-cap is equal to \(k=q^2+1\) and it is known that every \((q^2+1)\)-cap is an elliptic quadric. A well-known technique for constructing new examples of caps is to construct caps containing a large number of points of an elliptic quadric \(Q\) and some other points not belonging to this quadric \(Q\). Using this method, \(k\)-caps of cardinality approximately equal to \(q^2/2\) have been constructed. For instance, B. Segre constructed caps of size \((q^2+q+2)/2 \) and of size \((q^2+q+4)/2\), by using this method. In this article, the above mentioned method is studied in detail to construct complete \(k\)-caps in \(PG(3,q)\), \(q\) odd, of size \(k=(m+1)(q+1)+\omega\), where \(m\) is the greatest integer such that \(m(m-1)/2 \leq (q+1)/4\) and where \(\omega\) is some value belonging to \(\{0,1,2\}\). These \(k\)-caps are the smallest complete \(k\)-caps in \(PG(3,q)\), \(q\) odd, presently known. An alternative construction of complete \(k\)-caps, with \(k=4+(q-1)^2/2\), is also presented. Again, this is based on the idea of constructing a complete cap consisting of a large number of points of a given elliptic quadric, and some points not belonging to this given elliptic quadric.