33 research outputs found
Improved upper bounds for partial spreads
A partial -spread in is a collection of
-dimensional subspaces with trivial intersection, i.e., each point is
covered at most once. So far the maximum size of a partial -spread in
was known for the cases , and with the additional requirements and
. We completely resolve the case for the binary case
.Comment: 8 page
Upper bounds for partial spreads
A partial -spread in is a collection of -dimensional
subspaces with trivial intersection such that each non-zero vector is covered
at most once. We present some improved upper bounds on the maximum sizes.Comment: 4 page
Finite geometries: pure mathematics close to applications
The research field of finite geometries investigates structures with a finite number of objects. Classical examples include vector spaces, projective spaces, and affine spaces over finite fields. Although many of these structures are studied for their geometrical importance, they are also of great interest in other, more applied domains of mathematics. In this snapshot, finite vector spaces are introduced. We discuss the geometrical concept of partial t-spreads together with its implications for the “packing problem” and a recent application in the existence of “cooling codes”
On the lengths of divisible codes
In this article, the effective lengths of all -divisible linear codes
over with a non-negative integer are determined. For that
purpose, the -adic expansion of an integer is introduced. It is
shown that there exists a -divisible -linear code of
effective length if and only if the leading coefficient of the
-adic expansion of is non-negative. Furthermore, the maximum weight
of a -divisible code of effective length is at most ,
where denotes the cross-sum of the -adic expansion of .
This result has applications in Galois geometries. A recent theorem of
N{\u{a}}stase and Sissokho on the maximum size of a partial spread follows as a
corollary. Furthermore, we get an improvement of the Johnson bound for constant
dimension subspace codes.Comment: 17 pages, typos corrected; the paper was originally named "An
improvement of the Johnson bound for subspace codes
Johnson type bounds for mixed dimension subspace codes
Subspace codes, i.e., sets of subspaces of , are applied in
random linear network coding. Here we give improved upper bounds for their
cardinalities based on the Johnson bound for constant dimension codes.Comment: 16 pages, typos correcte
A new upper bound for subspace codes
It is shown that the maximum size of a binary subspace code of
packet length , minimum subspace distance , and constant dimension
is at most . In Finite Geometry terms, the maximum number of solids
in , mutually intersecting in at most a point, is at
most . Previously, the best known upper bound was
implied by the Johnson bound and the maximum size of partial
plane spreads in . The result was obtained by combining
the classification of subspace codes with parameters and
with integer linear programming techniques. The
classification of subspace codes is obtained as a
byproduct.Comment: 9 page