The Cyclic Model forPG(n, q) and a Construction of Arcs

The n -dimensional finite projective space, PG(n, q), admits a cyclic model, in which the set of points of PG(n, q) is identified with the elements of the group Zqn + qn − 1 + ⋯ + q + 1. It was proved by Hall (1974, Math. Centre Tracts, 57, 1–26) that in the cyclic model of PG(2, q), the additive inverse of a line is a conic. The following generalization of this result is proved: cyclic model of PG(n, q), the additive inverse of a line is a (q + 1)-arc if n + 1 is a prime and q + 1 > n. also shown that the additive inverse of a line is always a normal rational curve in some subspace PG(m, q), where m + 1| n + 1.