On cyclic caps in 4-dimensional projective spaces

Abstract

For any divisor k of q 4 − 1, the elements of a group of kth-roots of unity can be viewed as a cyclic point set C k in P G(4, q). An interesting problem, connected to the theory of BCH codes, is to determine the spectrum A(q) of maximal divisors k of q 4 − 1 for which C k is a cap. Recently, Bierbrauer and Edel [Edel and Bierbrauer (2004) Finite Fields Appl 10:168-182] have proved that 3(q 2 + 1) ∈ A(q) provided that q is an even non-square. In this paper, the odd order case is investigated. It is proved that the only integer m for which m(q 2 + 1) ∈ A(q) is m = 2 for q ≡ 3 (mod 4), m = 1 for q ≡ 1 (mod 4). It is also shown that when q ≡ 3 (mod 4), the cap C 2(q 2 +1) is complete.

Additive Patterns in Multiplicative Subgroups

“…For p = 19, by quadratic reciprocity (or directly), the degree of regularity of the graph G is r = For p = 17 the graph G is 2-regular, and is in fact a cycle: (1, 8, 13, 2, 16, 9, 4, 15) (note that all sums of adjacent elements in this cycle are quadratic residues). For p = 23 the set of quadratic residues is Q = {1, 2, 3, 4, 6, 8, 9, 12, 13, 16, 18}, the corresponding graph is 4-regular and it is not difficult to find in it a Hamilton cycle: (1, 2,16,8,4,9,3,6,18,13,12).…”

“…These are called cyclic caps in projective spaces, and have been studied extensively, see, for example, [30], [32], [12] and the references therein. In particular, by modifying the construction in [32] we can prove the following.…”