  In spite of a recent breakthrough on upper bounds of the size of cap sets (by
Croot, Lev and Pach (2017) and Ellenberg and Gijswijt (2017)), the classical
cap set constructions had not been affected. In this work, we introduce a very
different method of construction for caps in all affine spaces with odd prime
modulus $p$. Moreover, we show that for all primes $p \equiv 5 \bmod 6$ with $p
\leq 41$, the new construction leads to an exponentially larger growth of the
affine and projective caps in $\mathrm{AG}(n,p)$ and $\mathrm{PG}(n,p)$. For
example, when $p=23$, the existence of caps with growth $(8.0875\ldots)^n$
follows from a three-dimensional example of Bose (1947), and the only
improvement had been to $(8.0901\ldots)^n$ by Edel (2004), based on a
six-dimensional example. We improve this lower bound to $(9-o(1))^n$.
