258,612 research outputs found
On a generalization of distance sets
A subset in the -dimensional Euclidean space is called a -distance
set if there are exactly distinct distances between two distinct points in
and a subset is called a locally -distance set if for any point
in , there are at most distinct distances between and other points
in .
Delsarte, Goethals, and Seidel gave the Fisher type upper bound for the
cardinalities of -distance sets on a sphere in 1977. In the same way, we are
able to give the same bound for locally -distance sets on a sphere. In the
first part of this paper, we prove that if is a locally -distance set
attaining the Fisher type upper bound, then determining a weight function ,
is a tight weighted spherical -design. This result implies that
locally -distance sets attaining the Fisher type upper bound are
-distance sets. In the second part, we give a new absolute bound for the
cardinalities of -distance sets on a sphere. This upper bound is useful for
-distance sets for which the linear programming bound is not applicable. In
the third part, we discuss about locally two-distance sets in Euclidean spaces.
We give an upper bound for the cardinalities of locally two-distance sets in
Euclidean spaces. Moreover, we prove that the existence of a spherical
two-distance set in -space which attains the Fisher type upper bound is
equivalent to the existence of a locally two-distance set but not a
two-distance set in -space with more than points. We also
classify optimal (largest possible) locally two-distance sets for dimensions
less than eight. In addition, we determine the maximum cardinalities of locally
two-distance sets on a sphere for dimensions less than forty.Comment: 17 pages, 1 figur
Good Random Matrices over Finite Fields
The random matrix uniformly distributed over the set of all m-by-n matrices
over a finite field plays an important role in many branches of information
theory. In this paper a generalization of this random matrix, called k-good
random matrices, is studied. It is shown that a k-good random m-by-n matrix
with a distribution of minimum support size is uniformly distributed over a
maximum-rank-distance (MRD) code of minimum rank distance min{m,n}-k+1, and
vice versa. Further examples of k-good random matrices are derived from
homogeneous weights on matrix modules. Several applications of k-good random
matrices are given, establishing links with some well-known combinatorial
problems. Finally, the related combinatorial concept of a k-dense set of m-by-n
matrices is studied, identifying such sets as blocking sets with respect to
(m-k)-dimensional flats in a certain m-by-n matrix geometry and determining
their minimum size in special cases.Comment: 25 pages, publishe
The Merrifield-Simmons conjecture holds for bipartite graphs
Let be a graph and the number of independent
(vertex) sets in . Then the Merrifield-Simmons conjecture states that the
sign of the term only depends on the parity of the distance of the vertices
in . We prove that the conjecture holds for bipartite graphs by
considering a generalization of the term, where vertex subsets instead of
vertices are deleted.Comment: 8 page
Between Shapes, Using the Hausdorff Distance
Given two shapes A and B in the plane with Hausdorff distance 1, is there a shape S with Hausdorff distance 1/2 to and from A and B? The answer is always yes, and depending on convexity of A and/or B, S may be convex, connected, or disconnected. We show a generalization of this result on Hausdorff distances and middle shapes, and show some related properties. We also show that a generalization of such middle shapes implies a morph with a bounded rate of change. Finally, we explore a generalization of the concept of a Hausdorff middle to more than two sets and show how to approximate or compute it
Rate-distance tradeoff for codes above graph capacity
The capacity of a graph is defined as the rate of exponential growth of
independent sets in the strong powers of the graph. In the strong power an edge
connects two sequences if at each position their letters are equal or adjacent.
We consider a variation of the problem where edges in the power graphs are
removed between sequences which differ in more than a fraction of
coordinates. The proposed generalization can be interpreted as the problem of
determining the highest rate of zero undetected-error communication over a link
with adversarial noise, where only a fraction of symbols can be
perturbed and only some substitutions are allowed.
We derive lower bounds on achievable rates by combining graph homomorphisms
with a graph-theoretic generalization of the Gilbert-Varshamov bound. We then
give an upper bound, based on Delsarte's linear programming approach, which
combines Lov\'asz' theta function with the construction used by McEliece et al.
for bounding the minimum distance of codes in Hamming spaces.Comment: 5 pages. Presented at 2016 IEEE International Symposium on
Information Theor
- …