Abstract
In this note we investigate the sharpness of Bruen’s bound on the size of a t-fold blocking set in \(AG(n,q)\) with respect to the hyperplanes. We give a construction for t-fold blocking sets meeting Bruen’s bound with \(t=q-n+2\). This construction is used further to find the minimal size of a t-fold affine blocking set with \(t=q-n+1\). We prove that for blocking sets in the geometries \(AG(n,2)\) the difference between the size of an optimal t-fold blocking set and tn exceeds any given number. In particular, we deviate infinitely from Bruen’s bound as n goes to infinity. We conclude with a construction that gives t-fold blocking sets with \(t=q-n+3\) whose size is close to the lower bounds known so far.
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Acknowledgments
This research has been partially supported by the Scientific Research Fund of Sofia University under Contract No. 109/09.05.2012. The first author greatfully acknowledges the support obtained by the project “Finite geometries, coding theory and cryptography” between the Research Foundation—Flanders (FWO) and the Bulgarian Academy of Sciences. The authors would like to thank Simeon Ball for the helpful discussions.
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Communicated by S. Ball.
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Landjev, I., Rousseva, A. On the sharpness of Bruen’s bound for intersection sets in Desarguesian affine spaces. Des. Codes Cryptogr. 72, 551–558 (2014). https://doi.org/10.1007/s10623-012-9783-2
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DOI: https://doi.org/10.1007/s10623-012-9783-2