Abstract
In this paper, results of a non-exhaustive computer search for unitals in the known planes of order twenty-five are reported. The 2-(126, 6, 1) designs associated with newly found unitals are studied in detail. 938 non-isomorphic unital designs are discovered and we show that three of the unital designs are embeddable in two non-isomorphic planes and 239 of them are resolvable. The findings of this study improve some well-known lower bounds on the number of such designs and provide new connections between some pairs of planes. A conjecture concerning the p-ranks of unital designs embedded in planes of order \(q^2\) is formulated.
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Acknowledgements
The authors would like to thank Vladimir Tonchev for suggesting the problem of develo** an algorithm for finding unitals in projective planes and for giving the general idea of such algorithm (use of unions of combinations of orbits of different subgroups of the plane’s automorphism group), and for the extensive discussions and exchanges for many years. The authors would like to thank also the anonymous referee for reading carefully the manuscript and making several useful remarks.
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A Some Statistics Related to Unital Designs Embedded in Projective Planes of Order 25
A Some Statistics Related to Unital Designs Embedded in Projective Planes of Order 25
See Table 3.
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Stoichev, S.D., Gezek, M. Unitals in Projective Planes of Order 25. Math.Comput.Sci. 17, 5 (2023). https://doi.org/10.1007/s11786-023-00556-9
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DOI: https://doi.org/10.1007/s11786-023-00556-9