There is no anti-cocircular pair of ovoids in PG(3,8)
Year: 2026
Type: preprint
Abstract: Two ovoids O1, O2 of PG(3,q) form an anti-cocircular pair if there is a bijection φ: O1 → O2 such that no four coplanar points of O1 have coplanar images. Such pairs are the two-row covering perfect hash families CPHF(2; q^2+1, q, 4) of maximal length. They exist for q=4 and not for odd q ≥ 7, and Bai XunDi asked whether they exist for infinitely many even q, the case q=8 being open. We show that there is none for q=8: neither the elliptic quadric nor the Tits ovoid of PG(3,8) contains 8 points with no four coplanar, while the image of a plane section of O1 would be 9 such points in O2. The proof uses the classification of the ovoids of PG(3,8) and an exhaustive computation, done by two independent programs. For every q=2^h ≥ 8 we also prove that neither ovoid of an anti-cocircular pair can be an elliptic quadric or a Tits ovoid. With the known classifications of ovoids, this shows that there is no anti-cocircular pair for q=16, 32, 64 either. The archive contains the programs and the data behind the arc counts.
Author: Sitjar, Théo Nicolas
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225772825
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