A geometric construction of an optimal [67,9,30] binary code

We construct a new optimal binary [67,9,30]-code corresponding to a complete cap of PG(8,2). This cap is obtained by gluing together some Hermitian Veroneseans in a mixed partition of PG(8,2) and one hyperoval. We also construct a ternary [364,9,234]-code corresponding to a complete cap of PG(8,3). This cap turns out to be as large as the largest known cap in PG(8,3) (in an asymptotical sense).