Mixed partitions of PG(3,q2)

AbstractA mixed partition of PG(2n−1,q2) is a partition of the points of PG(2n−1,q2) into (n−1)-spaces and Baer subspaces of dimension 2n−1. In (Bruck and Bose, J. Algebra 1 (1964) 85) it is shown that such a mixed partition of PG(2n−1,q2) can be used to construct a (2n−1)-spread of PG(4n−1,q) and hence a translation plane of order q2n. In this paper, we provide several new examples of such mixed partitions in the case when n=2

Subgeometry partitions from cyclic semifields

New cyclic semifield planes of order qlcm(m,n) are constructed. By varying m and n, while preserving the lcm(m,n), necessarily mutually non-isomorphic semifield planes are obtained. If lcm(m,n)/m = 3, new GL(2,qm) - q3m-planes are constructed. If m is even, new subgeometry partitions in PG(lcm(m, n)-1, q2), by subgeometries isomorphic to either PG(lcm(m,n)/2-1, q2) or PG(lcm(m,n)-1, q) are constructed. If the 2-order of m is strictly larger than the 2-order of n then ‘double’ retraction is possible producing two distinct subgeometry partitions from the same semifield plane. If m is even and lcm(m,n)/m = 3, new subgeometry partitions may be constructed from the GL(2,qm) - q3m-planes

Geometry of the inversion in a finite field and partitions of PG(2k−1,q){\mathrm{PG}}(2^k-1,q) in normal rational curves

Let $L=\mathbb F_{q^n}$ be a finite field and let $F=\mathbb F_q$ be a subfield of $L$. Consider $L$ as a vector space over $F$ and the associated projective space that is isomorphic to ${\mathrm{PG}}(n-1,q)$. The properties of the projective mapping induced by $x\mapsto x^{-1}$ have been studied in \cite{Cs13,Fa02,Ha83,He85,Bu95}, where it is proved that the image of any line is a normal rational curve in some subspace. In this note a more detailed geometric description is achieved. Consequences are found related to mixed partitions of the projective spaces; in particular, it is proved that for any positive integer $k$, if $q\ge2^k-1$, then there are partitions of ${\mathrm{PG}}(2^k-1,q)$ in normal rational curves of degree $2^k-1$. For smaller $q$ the same construction gives partitions in $(q+1)$-tuples of independent points

Mixed Bundling Auctions

We study multi-object auctions where agents have private and additive valuations for heterogeneous objects. We focus on the revenue properties of a class of dominant strategy mechanisms where a weight is assigned to each partition of objects. The weights influence the probability with which partitions are chosen in the mechanism. This class contains efficient auctions, pure bundling auctions, mixed bundling auctions, auctions with reserve prices and auctions with pre-packaged bundles. For any number of objects and bidders, both the pure bundling auction and separate, efficient auctions for the single objects are revenue-inferior to an auction that involves mixed bundling

Mixed Bundling Auctions

We study multi-object auctions where agents have private and additive valuations for heterogeneous objects. We focus on the revenue properties of a class of dominant strategy mechanisms where a weight is assigned to each partition of objects. The weights influence the probability with which partitions are chosen in the mechanism. This class contains efficient auctions, pure bundling auctions, mixed bundling auctions, auctions with reserve prices and auctions with pre-packaged bundles. For any number of objects and bidders, both the pure bundling auction and separate, efficient auctions for the single objects are revenue-inferior to an auction that involves mixed bundling.

Subspace code constructions

We improve on the lower bound of the maximum number of planes of ${\rm PG}(8,q)$ mutually intersecting in at most one point leading to the following lower bound: ${\cal A}_q(9, 4; 3) \ge q^{12}+2q^8+2q^7+q^6+q^5+q^4+1$ for constant dimension subspace codes. We also construct two new non-equivalent $(6, (q^3-1)(q^2+q+1), 4; 3)_q$ constant dimension subspace orbit-codes