Construction of (k,r)-caps in the Projective 3-Space PG(3,q) over the Galois Field GF(19)

Abstract

The aim of this paper is to construct and investigate (k,r)-cap in the projective three-dimensional space PG(3,19) defined over the finite field GF(19). In particular, we present an explicit construction of the maximum complete cap-(k,2) in PG(3,19), commonly referred to as an ovoid, where . By systematically adjoining zero-index points, this construction is extended to obtain maximal complete (k,r)-caps for all . These results confirm both the existence and completeness of such configurations and provide explicit examples that may serve as a foundation for further developments in finite geometry and applications to coding theory.