Abstract
In \(\mathrm{PG}(2,q^3)\), let \(\pi \) be a subplane of order \(q\) that is tangent to \(\ell _\infty \). The tangent splash of \(\pi \) is defined to be the set of \(q^2+1\) points on \(\ell _\infty \) that lie on a line of \(\pi \). This article investigates properties of the tangent splash. We show that all tangent splashes are projectively equivalent, investigate sublines contained in a tangent splash, and consider the structure of a tangent splash in the Bruck–Bose representation of \(\mathrm{PG}(2,{q}^3)\) in \(\mathrm{PG}(6,{q})\). We show that a tangent splash of \(\mathrm{PG}(1,q^3)\) is a \(\mathrm{\text{ GF }}(q)\)-linear set of rank 3 and size \(q^2+1\); this allows us to use results about linear sets from Lavrauw and Van de Voorde (Des. Codes Cryptogr. 56:89–104, 2010) to obtain properties of tangent splashes.
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Acknowledgments
We would like to thank the anonymous referees. One for pointing out the important relationship between tangent splashes and linear sets, and that a number of our results could be proved more directly using the theory of linear sets. The other for helpful suggestions regarding Sect. 4, and for noting that the affine plane of Theorem 3.2 is Desarguesian.
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Communicated by J. D. Key.
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Barwick, S.G., Jackson, WA. An investigation of the tangent splash of a subplane of \(\mathrm{PG}(2,q^3)\) . Des. Codes Cryptogr. 76, 451–468 (2015). https://doi.org/10.1007/s10623-014-9971-3
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DOI: https://doi.org/10.1007/s10623-014-9971-3