Abstract
A t-intersecting constant dimension subspace code C is a set of k-dimensional subspaces in a projective space \(\mathrm {PG}(n,q)\), where distinct subspaces intersect in exactly a t-dimensional subspace. A classical example of such a code is the sunflower, where all subspaces pass through the same t-space. The sunflower bound states that such a code is a sunflower if \(|C| > \left( \frac{q^{k + 1} - q^{t + 1}}{q - 1} \right) ^2 + \left( \frac{q^{k + 1} - q^{t + 1}}{q - 1} \right) + 1\). In this article we will look at the case \(t=0\) and we will improve this bound for \(q\ge 9\): a set \(\mathcal {S}\) of k-spaces in \(\mathrm {PG}(n,q), q\ge 9\), pairwise intersecting in a point is a sunflower if \(|\mathcal {S}|> \left( \frac{2}{\root 6 \of {q}}+\frac{4}{\root 3 \of {q}}- \frac{5}{\sqrt{q}}\right) \left( \frac{q^{k + 1} - 1}{q - 1}\right) ^2\).
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Acknowledgements
The research of Jozefien D’haeseleer is supported by the FWO (Research Foundation Flanders). We would like to thank our colleague Lins Denaux for proof-reading this article in detail.
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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue: The Art of Combinatorics – A Volume in Honour of Aart Blokhuis”.
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Blokhuis, A., De Boeck, M. & D’haeseleer, J. On the sunflower bound for k-spaces, pairwise intersecting in a point. Des. Codes Cryptogr. 90, 2101–2111 (2022). https://doi.org/10.1007/s10623-021-00949-6
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DOI: https://doi.org/10.1007/s10623-021-00949-6