1,327 research outputs found
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
Freiman's theorem in finite fields via extremal set theory
Using various results from extremal set theory (interpreted in the language
of additive combinatorics), we prove an asyptotically sharp version of
Freiman's theorem in F_2^n: if A in F_2^n is a set for which |A + A| <= K|A|
then A is contained in a subspace of size 2^{2K + O(\sqrt{K}\log K)}|A|; except
for the O(\sqrt{K} \log K) error, this is best possible. If in addition we
assume that A is a downset, then we can also cover A by O(K^{46}) translates of
a coordinate subspace of size at most |A|, thereby verifying the so-called
polynomial Freiman-Ruzsa conjecture in this case. A common theme in the
arguments is the use of compression techniques. These have long been familiar
in extremal set theory, but have been used only rarely in the additive
combinatorics literature.Comment: 18 page
Families of nested completely regular codes and distance-regular graphs
In this paper infinite families of linear binary nested completely regular
codes are constructed. They have covering radius equal to or ,
and are -th parts, for of binary (respectively,
extended binary) Hamming codes of length (respectively, ), where
. In the usual way, i.e., as coset graphs, infinite families of embedded
distance-regular coset graphs of diameter equal to or are
constructed. In some cases, the constructed codes are also completely
transitive codes and the corresponding coset graphs are distance-transitive
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