534 research outputs found
Optimal Binary Subspace Codes of Length 6, Constant Dimension 3 and Minimum Distance 4
It is shown that the maximum size of a binary subspace code of packet length
, minimum subspace distance , and constant dimension is ;
in Finite Geometry terms, the maximum number of planes in
mutually intersecting in at most a point is .
Optimal binary subspace codes are classified into
isomorphism types, and a computer-free construction of one isomorphism type is
provided. The construction uses both geometry and finite fields theory and
generalizes to any , yielding a new family of -ary
subspace codes
On Projective Equivalence of Univariate Polynomial Subspaces
We pose and solve the equivalence problem for subspaces of ,
the dimensional vector space of univariate polynomials of degree . The group of interest is acting by projective transformations
on the Grassmannian variety of -dimensional
subspaces. We establish the equivariance of the Wronski map and use this map to
reduce the subspace equivalence problem to the equivalence problem for binary
forms
Problems on q-Analogs in Coding Theory
The interest in -analogs of codes and designs has been increased in the
last few years as a consequence of their new application in error-correction
for random network coding. There are many interesting theoretical, algebraic,
and combinatorial coding problems concerning these q-analogs which remained
unsolved. The first goal of this paper is to make a short summary of the large
amount of research which was done in the area mainly in the last few years and
to provide most of the relevant references. The second goal of this paper is to
present one hundred open questions and problems for future research, whose
solution will advance the knowledge in this area. The third goal of this paper
is to present and start some directions in solving some of these problems.Comment: arXiv admin note: text overlap with arXiv:0805.3528 by other author
Persistence for Circle Valued Maps
We study circle valued maps and consider the persistence of the homology of
their fibers. The outcome is a finite collection of computable invariants which
answer the basic questions on persistence and in addition encode the topology
of the source space and its relevant subspaces. Unlike persistence of real
valued maps, circle valued maps enjoy a different class of invariants called
Jordan cells in addition to bar codes. We establish a relation between the
homology of the source space and of its relevant subspaces with these
invariants and provide a new algorithm to compute these invariants from an
input matrix that encodes a circle valued map on an input simplicial complex.Comment: A complete algorithm to compute barcodes and Jordan cells is provided
in this version. The paper is accepted in in the journal Discrete &
Computational Geometry. arXiv admin note: text overlap with arXiv:1210.3092
by other author
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