Generalised dual arcs and Veronesean surfaces, with applications to cryptography

We start by defining generalised dual arcs, the motivation for defining them comes from cryptography, since they can serve as a tool to construct authentication codes and secret sharing schemes. We extend the characterisation of the tangent planes of the Veronesean surface V 2 4 in PG ( 5 , q ) , q odd, described in [J.W.P. Hirschfeld, J.A. Thas, General Galois Geometries, Oxford Math. Monogr., Clarendon Press/Oxford Univ. Press, New York, 1991], as a set of q 2 + q + 1 planes in PG ( 5 , q ) , such that every two intersect in a point and every three are skew. We show that a set of q 2 + q planes generating PG ( 5 , q ) , q odd, and satisfying the above properties can be extended to a set of q 2 + q + 1 planes still satisfying all conditions. This result is a natural generalisation of the fact that a q-arc in PG ( 2 , q ) , q odd, can always be extended to a ( q + 1 ) -arc. This extension result is then used to study a regular generalised dual arc with parameters ( 9 , 5 , 2 , 0 ) in PG ( 9 , q ) , q odd, where we obtain an algebraic characterisation of such an object as being the image of a cubic Veronesean.