Abstract We provide classification results for translation generalized quadrangles of order less than or equal to 64, and hence, for all incidence geometries related to them. The results consist of the classification of all pseudo-ovals in $$\textrm{PG}(3n-1,2)$$ PG ( 3 n - 1 , 2 ) , for $$n=3,4$$ n = 3 , 4 , and that of the pseudo-ovals in $$\textrm{PG}(3n-1,q)$$ PG ( 3 n - 1 , q ) , for $$n=5,6$$ n = 5 , 6 , such that one of the associated projective planes is Desarguesian.

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).