Small Complete Caps in Spaces of Even Characteristic

In 1959, Segre constructed a complete (3q+2)-cap inPG(3, q),qeven. This showed that the size of the smallest completek-cap inPG(3, q),qeven, is almost equal to the trivial lower bound which is of order[formula]. Generalizing the construction of Segre, complete (qn+3(qn−1+…+q)+2)-caps inPG(2n, q),qeven,q⩾4, and complete (3(qn+…+q)+2)-caps inPG(2n+1, q),qeven,q⩾4, are constructed. This shows that in all spacesPG(2n+1, q),qeven, the size of the smallest completek-cap is almost equal to the trivial lower bound which is of order[formula].