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Plane curves giving rise to blocking sets over finite fields

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Abstract

In recent years, many useful applications of the polynomial method have emerged in finite geometry. Indeed, algebraic curves, especially those defined by Rédei-type polynomials, are powerful in studying blocking sets. In this paper, we reverse the engine and study when blocking sets can arise from rational points on plane curves over finite fields. We show that irreducible curves of low degree cannot provide blocking sets and prove more refined results for cubic and quartic curves. On the other hand, using tools from number theory, we construct smooth plane curves defined over \(\mathbb {F}_p\) of degree at most \(4p^{3/4}+1\) whose points form blocking sets.

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Acknowledgements

The authors thank Greg Martin and József Solymosi for helpful discussions. During the preparation of this manuscript, the first author was supported by a postdoctoral research fellowship from the University of British Columbia and the NSERC PDF award. The second author is supported by an NSERC Discovery grant.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Santa Clara University, 500 El Camino Real, Santa Clara, 95053, USA

    Shamil Asgarli

  2. Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, V6T 1Z2, Canada

    Dragos Ghioca & Chi Hoi Yip

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  1. Shamil Asgarli
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  2. Dragos Ghioca
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  3. Chi Hoi Yip
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Correspondence to Chi Hoi Yip.

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Communicated by G. Korchmaros.

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Asgarli, S., Ghioca, D. & Yip, C.H. Plane curves giving rise to blocking sets over finite fields. Des. Codes Cryptogr. 91, 3643–3669 (2023). https://doi.org/10.1007/s10623-023-01264-y

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