Exponentially Larger Affine and Projective Caps

In spite of a recent breakthrough on upper bounds of the size of cap sets (by Croot, Lev and Pach and by Ellenberg and Gijswijt), 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 (Formula presented.) with (Formula presented.), the new construction leads to an exponentially larger growth of the affine and projective caps in (Formula presented.) and (Formula presented.). For example, when (Formula presented.), the existence of caps with growth (Formula presented.) follows from a three-dimensional example of Bose, and the only improvement had been to (Formula presented.) by Edel, based on a six-dimensional example. We improve this lower bound to (Formula presented.).