On \(\lambda \)-fold partitions of finite vector spaces and duality. (English) Zbl 1290.05017
Summary: Vector space partitions of an \(n\)-dimensional vector space \(V\) over a finite field are considered in [T. Bu, ibid. 31, 79–83 (1980; Zbl 0445.15004); O. Heden, Arch. Math. 43, 507–509 (1984; Zbl 0554.15004)], and more recently in a series of papers by A. D. Blinco et al. [Des. Codes Cryptography 48, No. 1, 69–77 (2008; Zbl 1184.15002)] and S. I. El-Zanati et al. [J. Comb. Des. 16, No. 4, 329–341 (2008; Zbl 1176.05018); Discrete Math. 309, No. 14, 4727–4735 (2009; Zbl 1269.15001)].
In this paper, we consider the generalization of a vector space partition which we call a \(\lambda \)-fold partition (or simply a \(\lambda \)-partition). In particular, for a given positive integer, \(\lambda \), we define a \(\lambda \)-fold partition of \(V\) to be a multiset of subspaces of \(V\) such that every nonzero vector in \(V\) is contained in exactly \(\lambda \) subspaces in the given multiset. A \(\lambda \)-fold spread as defined in [J. Hirschfeld, Projective geometries over finite fields. 2nd ed. Oxford: Clarendon Press (1998; Zbl 0899.51002)] is one example of a \(\lambda \)-fold partition. After establishing some definitions in the introduction, we state some necessary conditions for a \(\lambda \)-fold partition of \(V\) to exist, then introduce some general ways to construct such partitions.
We also introduce the construction of a dual \(\lambda \)-partition as a way of generating \(\lambda'\)-partitions from a given \(\lambda \)-partition. One application of this construction is that the dual of a vector space partition will, in general, be a \(\lambda \)-partition for some \(\lambda >1\). In the last section, we discuss a connection between \(\lambda \)-partitions and some designs over finite fields.
In this paper, we consider the generalization of a vector space partition which we call a \(\lambda \)-fold partition (or simply a \(\lambda \)-partition). In particular, for a given positive integer, \(\lambda \), we define a \(\lambda \)-fold partition of \(V\) to be a multiset of subspaces of \(V\) such that every nonzero vector in \(V\) is contained in exactly \(\lambda \) subspaces in the given multiset. A \(\lambda \)-fold spread as defined in [J. Hirschfeld, Projective geometries over finite fields. 2nd ed. Oxford: Clarendon Press (1998; Zbl 0899.51002)] is one example of a \(\lambda \)-fold partition. After establishing some definitions in the introduction, we state some necessary conditions for a \(\lambda \)-fold partition of \(V\) to exist, then introduce some general ways to construct such partitions.
We also introduce the construction of a dual \(\lambda \)-partition as a way of generating \(\lambda'\)-partitions from a given \(\lambda \)-partition. One application of this construction is that the dual of a vector space partition will, in general, be a \(\lambda \)-partition for some \(\lambda >1\). In the last section, we discuss a connection between \(\lambda \)-partitions and some designs over finite fields.
MSC:
| 05A18 | Partitions of sets |
| 05B30 | Other designs, configurations |
| 05B05 | Combinatorial aspects of block designs |
| 05B25 | Combinatorial aspects of finite geometries |
| 15A03 | Vector spaces, linear dependence, rank, lineability |
Citations:
Zbl 0445.15004; Zbl 0554.15004; Zbl 1184.15002; Zbl 1176.05018; Zbl 1269.15001; Zbl 0899.51002References:
| [1] | Assmus, E. F.; Key, J. D., Designs and their Codes (1993), Cambridge University Press: Cambridge University Press Cambridge, UK · Zbl 0762.05001 |
| [2] | Beth, T.; Jungnickel, D.; Lenz, H., (Design Theory. Vol. I. Design Theory. Vol. I, Encyclopedia Math. Appl., vol. 78 (1999), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 0945.05005 |
| [3] | Blinco, A. D.; El-Zanati, S. I.; Seelinger, G. F.; Sissokho, P. A.; Spence, L. E.; Vanden Eynden, C., On vector space partitions and uniformly resolvable designs, Des. Codes Cryptogr., 15, 6, 69-77 (2008) · Zbl 1184.15002 |
| [4] | Braun, M.; Kerber, A.; Laue, R., Systematic construction of \(q\)-analogs of designs, Des. Codes Cryptogr., 34, 55-70 (2005) · Zbl 1055.05009 |
| [5] | Bu, T., Partitions of a vector space, Discrete Math., 31, 79-83 (1980) · Zbl 0445.15004 |
| [6] | Curtis, C. W., Linear Algebra. An Introductory Approach (1984), Springer-Verlag: Springer-Verlag New York · Zbl 0159.31901 |
| [7] | Danziger, P.; Rodney, P., Uniformly resolvable designs, (Colbourn, Charles J.; Dinitz, Jeffrey H., The CRC Handbook of Combinatorial Designs. The CRC Handbook of Combinatorial Designs, CRC Press Series on Discrete Mathematics and its Applications (1996), CRC Press: CRC Press Boca Raton, FL), 490-492 · Zbl 0847.05009 |
| [8] | El-Zanati, S. I.; Seelinger, G. F.; Sissokho, P. A.; Spence, L. E.; Vanden Eynden, C., Partitions of finite vector spaces into subspaces, J. Combin. Des., 16, 4, 329-341 (2008) · Zbl 1176.05018 |
| [9] | El-Zanati, S. I.; Seelinger, G. F.; Sissokho, P. A.; Spence, L. E.; Vanden Eynden, C., On partitions of finite vector spaces of low dimension over \(G F(2)\), Discrete Math., 309, 4727-4735 (2009) · Zbl 1269.15001 |
| [10] | Heden, O., On partitions of finite vector spaces of small dimension, Arch. Math., 43, 507-509 (1984) · Zbl 0554.15004 |
| [11] | Heden, O., Partitions of finite abelian groups, European J. Combin., 7, 11-25 (1986) · Zbl 0651.20035 |
| [12] | Hirschfeld, J., Projective Geometries Over Finite Fields (1998), Oxford University Press: Oxford University Press Oxford, UK · Zbl 0899.51002 |
| [13] | Thomas, S., Designs over finite fields, Geom. Dedicata, 24, 237-242 (1987) · Zbl 0627.51013 |
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