10,254 research outputs found
X-code: MDS array codes with optimal encoding
We present a new class of MDS (maximum distance separable) array codes of size n×n (n a prime number) called X-code. The X-codes are of minimum column distance 3, namely, they can correct either one column error or two column erasures. The key novelty in X-code is that it has a simple geometrical construction which achieves encoding/update optimal complexity, i.e., a change of any single information bit affects exactly two parity bits. The key idea in our constructions is that all parity symbols are placed in rows rather than columns
The weight hierarchies and chain condition of a class of codes from varieties over finite fields
The generalized Hamming weights of linear codes were first introduced by Wei. These are fundamental parameters related to the minimal overlap structures of the subcodes and very useful in several fields. It was found that the chain condition of a linear code is convenient in studying the generalized Hamming weights of the product codes. In this paper we consider a class of codes defined over some varieties in projective spaces over finite fields, whose generalized Hamming weights can be determined by studying the orbits of subspaces of the projective spaces under the actions of classical groups over finite fields, i.e., the symplectic groups, the unitary groups and orthogonal groups. We give the weight hierarchies and generalized weight spectra of the codes from Hermitian varieties and prove that the codes satisfy the chain condition
A semidefinite programming hierarchy for packing problems in discrete geometry
Packing problems in discrete geometry can be modeled as finding independent
sets in infinite graphs where one is interested in independent sets which are
as large as possible. For finite graphs one popular way to compute upper bounds
for the maximal size of an independent set is to use Lasserre's semidefinite
programming hierarchy. We generalize this approach to infinite graphs. For this
we introduce topological packing graphs as an abstraction for infinite graphs
coming from packing problems in discrete geometry. We show that our hierarchy
converges to the independence number.Comment: (v2) 25 pages, revision based on suggestions by referee, accepted in
Mathematical Programming Series B special issue on polynomial optimizatio
On codes satisfying the double chain condition
AbstractThe double chain condition is described. A number of bounds on the length and weight hierarchy of codes satisfying the double chain condition are given. Constructions of codes satisfying the double chain condition and with trellis complexity 1 or 2 are given
Communications and information research: Improved space link performance via concatenated forward error correction coding
With the development of new advanced instruments for remote sensing applications, sensor data will be generated at a rate that not only requires increased onboard processing and storage capability, but imposes demands on the space to ground communication link and ground data management-communication system. Data compression and error control codes provide viable means to alleviate these demands. Two types of data compression have been studied by many researchers in the area of information theory: a lossless technique that guarantees full reconstruction of the data, and a lossy technique which generally gives higher data compaction ratio but incurs some distortion in the reconstructed data. To satisfy the many science disciplines which NASA supports, lossless data compression becomes a primary focus for the technology development. While transmitting the data obtained by any lossless data compression, it is very important to use some error-control code. For a long time, convolutional codes have been widely used in satellite telecommunications. To more efficiently transform the data obtained by the Rice algorithm, it is required to meet the a posteriori probability (APP) for each decoded bit. A relevant algorithm for this purpose has been proposed which minimizes the bit error probability in the decoding linear block and convolutional codes and meets the APP for each decoded bit. However, recent results on iterative decoding of 'Turbo codes', turn conventional wisdom on its head and suggest fundamentally new techniques. During the past several months of this research, the following approaches have been developed: (1) a new lossless data compression algorithm, which is much better than the extended Rice algorithm for various types of sensor data, (2) a new approach to determine the generalized Hamming weights of the algebraic-geometric codes defined by a large class of curves in high-dimensional spaces, (3) some efficient improved geometric Goppa codes for disk memory systems and high-speed mass memory systems, and (4) a tree based approach for data compression using dynamic programming
Bond dimension witnesses and the structure of homogeneous matrix product states
For the past twenty years, Matrix Product States (MPS) have been widely used
in solid state physics to approximate the ground state of one-dimensional spin
chains. In this paper, we study homogeneous MPS (hMPS), or MPS constructed via
site-independent tensors and a boundary condition. Exploiting a connection with
the theory of matrix algebras, we derive two structural properties shared by
all hMPS, namely: a) there exist local operators which annihilate all hMPS of a
given bond dimension; and b) there exist local operators which, when applied
over any hMPS of a given bond dimension, decouple (cut) the particles where
they act from the spin chain while at the same time join (glue) the two loose
ends back again into a hMPS. Armed with these tools, we show how to
systematically derive `bond dimension witnesses', or 2-local operators whose
expectation value allows us to lower bound the bond dimension of the underlying
hMPS. We extend some of these results to the ansatz of Projected Entangled
Pairs States (PEPS). As a bonus, we use our insight on the structure of hMPS
to: a) derive some theoretical limitations on the use of hMPS and hPEPS for
ground state energy computations; b) show how to decrease the complexity and
boost the speed of convergence of the semidefinite programming hierarchies
described in [Phys. Rev. Lett. 115, 020501 (2015)] for the characterization of
finite-dimensional quantum correlations.Comment: Accepted for publication in Quantum. We still do not acknowledge
support from the European Research Counci
Lipschitz and uniformly continuous reducibilities on ultrametric Polish spaces
We analyze the reducibilities induced by, respectively, uniformly continuous,
Lipschitz, and nonexpansive functions on arbitrary ultrametric Polish spaces,
and determine whether under suitable set-theoretical assumptions the induced
degree-structures are well-behaved.Comment: 37 pages, 2 figures, revised version, accepted for publication in the
Festschrift that will be published on the occasion of Victor Selivanov's 60th
birthday by Ontos-Verlag. A mistake has been corrected in Section
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