A cap set in projective or affine geometry over a finite field is a set of points no three of which are collinear. In this paper, we propose a new construction for complete cap sets that yields a cap set of size 124928 in the affine geometry AG(15,3). It should be noted that the constructed cap set in AG(15,3) is more powerful and exceeds at least by 4096 points than those that can be obtained from the previously known ones using the product or doubling constructions.