Designs, Codes and Cryptography

, Volume 44, Issue 1–3, pp 15–23 | Cite as

Mixed partitions and related designs



We define a mixed partition of Π =  PG(d, q r ) to be a partition of the points of Π into subspaces of two distinct types; for instance, a partition of PG(2n − 1, q 2) into (n − 1)-spaces and Baer subspaces of dimension 2n − 1. In this paper, we provide a group theoretic method for constructing a robust class of such partitions. It is known that a mixed partition of PG(2n − 1, q 2) can be used to construct a (2n − 1)-spread of PG(4n − 1, q) and, hence, a translation plane of order q 2n . Here we show that our partitions can be used to construct generalized Andrè planes, thereby providing a geometric representation of an infinite family of generalized Andrè planes. The results are then extended to produce mixed partitions of PG(rn − 1, q r ) for r ≥ 3, which lift to (rn − 1)-spreads of PG(r 2 n − 1, q) and hence produce \(2-(q^{r^2n},q^{rn},1)\) (translation) designs with parallelism. These designs are not isomorphic to the designs obtained from the points and lines of AG(r, q rn ).


Mixed partitions Spreads Generalized Andre planes Singer groups Translation designs 

AMS Classification

51E20 51A35 


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Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of DelawareNewarkUSA
  2. 2.Department of MathematicsUniversity of Mary WashingtonFredericksburgUSA

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