Complete ( q 2?+ q +?8)/2-caps in the spaces PG (3, q ), q ≡ 2 (mod 3) an odd prime, and a complete 20-cap in PG (3, 5) |
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Authors: | Alexander A Davydov Stefano Marcugini and Fernanda Pambianco |
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Institution: | (1) Institute for Information Transmission Problems, Russian Academy of Sciences, Bol’shoi Karetnyi per. 19, GSP-4, Moscow, 127994, Russian Federation;(2) Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Via Vanvitelli 1, Perugia, 06123, Italy |
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Abstract: | An infinite family of complete (q
2 + q + 8)/2-caps is constructed in PG(3, q) where q is an odd prime ≡ 2 (mod 3), q ≥ 11. This yields a new lower bound on the second largest size of complete caps. A variant of our construction also produces
one of the two previously known complete 20-caps in PG(3, 5). The associated code weight distribution and other combinatorial properties of the new (q
2 + q + 8)/2-caps and the 20-cap in PG(3, 5) are investigated. The updated table of the known sizes of the complete caps in PG(3, q) is given. As a byproduct, we have found that the unique complete 14-arc in PG(2, 17) contains 10 points on a conic. Actually, this shows that an earlier general result dating back to the Seventies fails
for q = 17.
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Keywords: | Complete caps Projective spaces of the dimension three Projective planes |
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