On the spectrum of the sizes of maximal partial line spreads in \(\mathrm{PG}(2n,q)\), \(n \geq 3\). (English) Zbl 1074.51003
Summary: A lot of research has been done on the spectrum of the sizes of maximal partial spreads in \(\mathrm{PG}(3,q)\) [P. Govaerts and L. Storme, Des. Codes Cryptography 28, 51–63 (2003; Zbl 1022.51004); O. Heden, Discrete Math. 120, No. 1–3, 75–91 (1993; Zbl 0784.51007); ibid. 142, No. 1–3, 97–106 (1995; Zbl 0831.51006), and ibid. 243, 135–150 (2002; Zbl 0993.51002)]. In [A. Gács and T. Szőnyi, Des. Codes Cryptography 29, No. 1–3, 123–129 (2003; Zbl 1038.51005)], results on the spectrum of the sizes of maximal partial line spreads in \(\mathrm{PG}(N,q)\), \(N \geq 5\), are given.
In \(\mathrm{PG}(2n,q)\), \(n\geq 3\), the largest possible size for a partial line spread is \(q^{2n-1} + q^{2n-3} + \cdots + q^{3}+1\). The largest size for the maximal partial line spreads constructed by A. Gács and T. Szőnyi, [loc. cit.] is \((q^{2n+1} - q)/(q^{2}-1) - q^{3} + q^{2} - 2q +2\). This shows that there is a non-empty interval of values of \(k\) for which it is still not known whether there exists a maximal partial line spread of size \(k\) in \(\mathrm{PG}(2n,q)\).
We now show that there indeed exists a maximal partial line spread of size \(k\) for every value of \(k\) in that interval when \(q\geq 9\).
In \(\mathrm{PG}(2n,q)\), \(n\geq 3\), the largest possible size for a partial line spread is \(q^{2n-1} + q^{2n-3} + \cdots + q^{3}+1\). The largest size for the maximal partial line spreads constructed by A. Gács and T. Szőnyi, [loc. cit.] is \((q^{2n+1} - q)/(q^{2}-1) - q^{3} + q^{2} - 2q +2\). This shows that there is a non-empty interval of values of \(k\) for which it is still not known whether there exists a maximal partial line spread of size \(k\) in \(\mathrm{PG}(2n,q)\).
We now show that there indeed exists a maximal partial line spread of size \(k\) for every value of \(k\) in that interval when \(q\geq 9\).
MSC:
| 51E14 | Finite partial geometries (general), nets, partial spreads |
| 51E20 | Combinatorial structures in finite projective spaces |
References:
| [9] | Govaerts P., Heden O. and Storme L. On the spectrum of the size of maximal partial spreads in PG(3,q), q even. (In preparation). |
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