Mixed partitions and related designs

Autor: Gary L. Ebert, Keith E. Mellinger
Rok vydání: 2007
Předmět:
Zdroj: Designs, Codes and Cryptography. 44:15-23
ISSN: 1573-7586
0925-1022
DOI: 10.1007/s10623-007-9044-y
Popis: We define a mixed partition of ? = PG(d, q r ) to be a partition of the points of ? 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 Andre planes, thereby providing a geometric representation of an infinite family of generalized Andre 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 ).
Databáze: OpenAIRE
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