Some families of complete caps of quadrics and symplectic polarities over GF(8)

A cap on a quadric is a set of its points whose pairwise joins are all chords. A cap is complete if it is not part of a larger one. The only field for which all complete quadric caps are known is GF(2). Those caps are small; the biggest for each quadric is of order the dimension of the ambient space. Apart from information about ovoids in dimensions at most 7, little else is known. Here, the evidence is increased by providing caps over GF(2), odd, which, if >1, have size of order the dimension cubed. In particular, complete caps are obtained for the quadrics Q _2m (8), Q ₊ ^8k+7 (8), Q _- ^8k+3 (8), Q ₊ ^8k+1 (8) and Q _- ^8k+5 (8). These caps on Q ₊ ^8k+7 (8) and Q _- ^8k+3 (8) are complete on any Q _n(8) of which their quadrics are sections; so is that that of Q ₄₊₂(8) for any Q _2n (8) of which Q ₄₊₂(8) is a section with the same kernel. From the correspondence with Q _2n (8) complete caps are obtained for symplectic polarities over GF(8).