Abstract
We improve on the lower bound of the maximum number of planes in \({\text {PG}}(8,q)\cong \mathbb {F}_q^{9}\) pairwise intersecting in at most a point. In terms of constant dimension codes this leads to \(A_q(9,4;3)\ge q^{12}+ 2q^8+2q^7+q^6+2q^5+2q^4-2q^2-2q+1\). This result is obtained via a more general construction strategy, which also yields other improvements.
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Notes
The same applies to \(\mathcal {C}_1\), i.e., we can avoid to use [2, Theorem 3.12], see the subsequent Footnote 3.
Using linearized polynomials to described the lifted MRD code, a clique of matching size can be described as the set of monomials ax (including the zero polynomial).
Both constructions are stated in the language of linearized polynomials. For [9, Lemma 12, Example 4] the representation \(\mathbb {F}_q^6\cong \mathbb {F}_{q^3}\times \mathbb {F}_{q^3}\) is used and the planes removed from the lifted MRD code correspond to \(ux^q-u^qx\) for \(u\in \mathbb {F}_{q^3}\), so that the monomials ax for \(a\in \mathbb {F}_{q^3}\backslash \{\mathbf {0}\}\) correspond to a clique of cardinality \(q^3-1\). For [8, Theorem 4] the representation \(\mathbb {F}_q^7\cong W\times \mathbb {F}_{q^4}\), where W denotes the trace-zero subspace of \(\mathbb {F}_{q^4}/\mathbb {F}_q\), is used. The planes removed from the lifted MRD code correspond to \(r\left( ux^q-u^qx\right) \) for \(r\in \mathbb {F}_{q^4}\backslash \{\mathbf {0}\}\) and \(u\in \mathbb {F}_{q^4}\) with \({\text {tr}}(u)=1\), so that the monomial s ax for \(a\in \mathbb {F}_{q^4}\) correspond to a clique of cardinality \(q^4\).
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Acknowledgements
The author would like to thank Thomas Honold for his analysis of possible clique sizes in the constant dimension codes from [9, Lemma 12, Example 4] and [8, Theorem 4], see Footnote 3. The main idea for Theorem 3 is inspired by [2]. Further thanks go to the anonymous referees for their careful reading and helpful remarks.
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Communicated by G. Lunardon.
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Kurz, S. Subspaces intersecting in at most a point. Des. Codes Cryptogr. 88, 595–599 (2020). https://doi.org/10.1007/s10623-019-00699-6
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DOI: https://doi.org/10.1007/s10623-019-00699-6