Bicovering arcs and small complete caps from elliptic curves

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 ka parts per thousand currency signq/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 Na parts per thousand 0(mod4). When q=p (h) with p prime and ha parts per thousand currency sign8, these caps are the smallest known complete caps in AG(N,q), Na parts per thousand 0(mod4).