Bicovering arcs and small complete caps from elliptic curves

Abstract

Bicovering arcs in Galois affine planes of odd order are a powerful tool for the construction of complete caps in spaces of arbitrarily higher dimensions. The aim of this paper is to investigate whether the arcs contained in elliptic cubic curves are bicovering. As a result, bicovering k-arcs in AG(2,q) of size k≤q/3 are obtained, provided that q−1 has a prime divisor m with 7<m<(1/8)q 1/4. Such arcs produce complete caps of size kq (N−2)/2 in affine spaces of dimension N≡0(mod4). When q=p h with p prime and h≤8, these caps are the smallest known complete caps in AG(N,q), N≡0(mod4).