## Product of Two Caps with Tangent Hyperplane

Let C1 be an (s1 + 1)-cap in the projective space PG(u1, b) and C2 be an (s2 + 1)-cap in PG(u2, b), such that both of them possess a tangent hyperplane. In other words, there are points x1C1 and x2C2 such that C1 ∖ {x1} is an s1-cap the affine space AG(u1, b) and C2 ∖ {x2} is an s2-cap in AG(u2, b). Then it is shown in [1, Theorem 10] that an (s1s2 + s1 + s2)-cap in PG(u1 + u2, b) can be constructed.

The most common application of this method is to use ovoids as C1 as well as C2. The resulting (b4 +2b2)-caps in PG(6, b) (cf. [1, Theorem 11] and [2, Table 4.6(i)]) are the largest known caps in the 6-dimensional projective space.

If the resulting cap has a dimension u ≠ 6, consult the description of this method in order to find the proper affine caps.

### Construction

Let

C1 = {,…,,}

with ai, aFbu1, and

C2 = {,…,,}

with bj, bFbu2. Then the new cap consists of the disjoint union of the following three sets:

{ | 1 ≤ is1, 1 ≤ js2},
{ | 1 ≤ is1},

and

{ | 1 ≤ js2},

with cardinality s1s2, s1, and s2, respectively.