# On vector space partitions and uniformly resolvable designs

- 66 Downloads
- 13 Citations

## Abstract

Let *V* _{ n }(*q*) denote a vector space of dimension *n* over the field with *q* elements. A set \({\mathcal{P}}\) of subspaces of *V* _{ n }(*q*) is a *partition* of *V* _{ n }(*q*) if every nonzero vector in *V* _{ n }(*q*) is contained in exactly one subspace in \({\mathcal{P}}\). A *uniformly resolvable design* is a pairwise balanced design whose blocks can be resolved in such a way that all blocks in a given parallel class have the same size. A partition of *V* _{ n }(*q*) containing *a* _{ i } subspaces of dimension *n* _{ i } for 1 ≤ *i* ≤ *k* induces a uniformly resolvable design on *q* ^{ n } points with *a* _{ i } parallel classes with block size \(q^{n_i}\), 1 ≤ *i* ≤ *k*, and also corresponds to a factorization of the complete graph \(K_{q^n}\) into \(a_i K_{q^{n_i}}\) -factors, 1 ≤ *i* ≤ *k*. We present some sufficient and some necessary conditions for the existence of certain vector space partitions. For the partitions that are shown to exist, we give the corresponding uniformly resolvable designs. We also show that there exist uniformly resolvable designs on *q* ^{ n } points where corresponding partitions of *V* _{ n }(*q*) do not exist.

## Keywords

Vector space partitions Uniformly resolvable designs## AMS Classifications

05B05 05B25 15A03## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Beutelspacher A. (1978). Partitions of finite vector spaces: an application of the Frobenius number in geometry. Arch. Math. 31, 202–208MATHCrossRefMathSciNetGoogle Scholar
- 2.Bu T. (1980). Partitions of a vector space. Discrete Math. 31, 79–83MATHCrossRefMathSciNetGoogle Scholar
- 3.Clark W., Dunning L. (1992). Partial partitions of vector spaces arising from the construction of byte error control codes. Ars Combin. 33, 161–177MATHMathSciNetGoogle Scholar
- 4.Danziger P., Rodney P. (1996). Uniformly resolvable designs. In: Colbourn C.J., Dinitz J.H. (eds) The CRC Handbook of Combinatorial Designs. CRC Press Series on Discrete Mathematics and its Applications. CRC Press, Boca Raton, pp. 490–492Google Scholar
- 5.El-Zanati S.I., Seelinger G.F., Sissokho P.A., Spence L.E., Vanden Eynden C.: On partitions of finite vector spaces of small dimension over
*GF*(2). Discrete Math. (to appear).Google Scholar - 6.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. Design (to appear).Google Scholar
- 7.Heden O. (1984). On partitions of finite vector spaces of small dimensions. Arch. Math. 43, 507–509MATHCrossRefMathSciNetGoogle Scholar
- 8.Heden O. (1986). Partitions of finite abelian groups. Eur. J. Combin. 7, 11–25MATHMathSciNetGoogle Scholar
- 9.Rees R. (1987). Uniformly resolvable pairwise balanced designs with block sizes two and three. J. Comb. Theory Ser. A 45, 207–225MATHCrossRefMathSciNetGoogle Scholar
- 10.Shen H. (1996). Existence of resolvable group divisible designs with block size four and group size two or three. J. Shanghai Jiaotong Univ. (Engl. Ed.) 1, 68–70MathSciNetGoogle Scholar
- 11.Tannenbaum P. (1983). Partitions of \({\mathbb{Z}}_2^n\). SIAM J. Alg. Disc. 4, 22–29MATHCrossRefMathSciNetGoogle Scholar