Abstract
For any divisor k of q 4−1, the elements of a group of k th-roots of unity can be viewed as a cyclic point set C k in PG(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.
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Giulietti, M. On cyclic caps in 4-dimensional projective spaces. Des. Codes Cryptogr. 47, 135–143 (2008). https://doi.org/10.1007/s10623-007-9106-1
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DOI: https://doi.org/10.1007/s10623-007-9106-1