216,118 research outputs found
On -covering designs
A -covering design , , is a collection
of -spaces of such that every
-space of is contained in at least one element of
. Let denote the minimum number of
-spaces in a -covering design . In this paper
improved upper bounds on , ,
, , and , , are presented. The results are achieved by constructing the related
-covering designs
Covering of Subspaces by Subspaces
Lower and upper bounds on the size of a covering of subspaces in the
Grassmann graph \cG_q(n,r) by subspaces from the Grassmann graph
\cG_q(n,k), , are discussed. The problem is of interest from four
points of view: coding theory, combinatorial designs, -analogs, and
projective geometry. In particular we examine coverings based on lifted maximum
rank distance codes, combined with spreads and a recursive construction. New
constructions are given for with or . We discuss the density
for some of these coverings. Tables for the best known coverings, for and
, are presented. We present some questions concerning
possible constructions of new coverings of smaller size.Comment: arXiv admin note: text overlap with arXiv:0805.352
Characterization of intersecting families of maximum size in PSL(2,q)
We consider the action of the 2-dimensional projective special linear group PSL(2,q) on the projective line PG(1,q) over the finite field Fq, where q is an odd prime power. A subset S of PSL(2,q) is said to be an intersecting family if for any g1,g2∈S, there exists an element x∈PG(1,q) such that xg1=xg2. It is known that the maximum size of an intersecting family in PSL(2,q) is q(q−1)/2. We prove that all intersecting families of maximum size are cosets of point stabilizers for all odd prime powers q\u3e3
Large weight code words in projective space codes
AbstractRecently, a large number of results have appeared on the small weights of the (dual) linear codes arising from finite projective spaces. We now focus on the large weights of these linear codes. For q even, this study for the code Ck(n,q)⊥ reduces to the theory of minimal blocking sets with respect to the k-spaces of PG(n,q), odd-blocking the k-spaces. For q odd, in a lot of cases, the maximum weight of the code Ck(n,q)⊥ is equal to qn+⋯+q+1, but some unexpected exceptions arise to this result. In particular, the maximum weight of the code C1(n,3)⊥ turns out to be 3n+3n-1. In general, the problem of whether the maximum weight of the code Ck(n,q)⊥, with q=3h (h⩾1), is equal to qn+⋯+q+1, reduces to the problem of the existence of sets of points in PG(n,q) intersecting every k-space in 2(mod3) points
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