861 research outputs found
Speeding up Deciphering by Hypergraph Ordering
The "Gluing Algorithm" of Semaev [Des.\ Codes Cryptogr.\ 49 (2008), 47--60]
--- that finds all solutions of a sparse system of linear equations over the
Galois field --- has average running time where is the total number of
equations, and is the set of all unknowns actively
occurring in the first equations. Our goal here is to minimize the exponent
of in the case where every equation contains at most three unknowns.
%Applying hypergraph-theoretic methods we prove The main result states that if
the total number of unknowns is equal
to , then the best achievable exponent is between and for some
positive constants and $c_2.
50 Years of the Golomb--Welch Conjecture
Since 1968, when the Golomb--Welch conjecture was raised, it has become the
main motive power behind the progress in the area of the perfect Lee codes.
Although there is a vast literature on the topic and it is widely believed to
be true, this conjecture is far from being solved. In this paper, we provide a
survey of papers on the Golomb--Welch conjecture. Further, new results on
Golomb--Welch conjecture dealing with perfect Lee codes of large radii are
presented. Algebraic ways of tackling the conjecture in the future are
discussed as well. Finally, a brief survey of research inspired by the
conjecture is given.Comment: 28 pages, 2 figure
Diameter Perfect Lee Codes
Lee codes have been intensively studied for more than 40 years. Interest in
these codes has been triggered by the Golomb-Welch conjecture on the existence
of the perfect error-correcting Lee codes. In this paper we deal with the
existence and enumeration of diameter perfect Lee codes. As main results we
determine all for which there exists a linear diameter-4 perfect Lee code
of word length over and prove that for each there are
uncountable many diameter-4 perfect Lee codes of word length over This
is in a strict contrast with perfect error-correcting Lee codes of word length
over \ as there is a unique such code for and its is
conjectured that this is always the case when is a prime. We produce
diameter perfect Lee codes by an algebraic construction that is based on a
group homomorphism. This will allow us to design an efficient algorithm for
their decoding. We hope that this construction will turn out to be useful far
beyond the scope of this paper
A Combinatorial Problem Related to Sparse Systems of Equations
Nowadays sparse systems of equations occur frequently in science and
engineering. In this contribution we deal with sparse systems common in
cryptanalysis. Given a cipher system, one converts it into a system of sparse
equations, and then the system is solved to retrieve either a key or a
plaintext. Raddum and Semaev proposed new methods for solving such sparse
systems. It turns out that a combinatorial MaxMinMax problem provides bounds on
the average computational complexity of sparse systems. In this paper we
initiate a study of a linear algebra variation of this MaxMinMax problem
Spatial diffusion in a periodic optical lattice: revisiting the Sisyphus effect
We numerically study the spatial diffusion of an atomic cloud experiencing
Sisyphus cooling in a three-dimensional linlin optical lattice in a broad
range of lattice parameters. In particular, we investigate the dependence on
the size of the lattice sites which changes with the angle between the laser
beams. We show that the steady-state temperature is largely independent of the
lattice angle, but that the spatial diffusion changes significantly. It is
shown that the numerical results fulfil the Einstein relations of Brownian
motion in the jumping regime as well as in the oscillating regime. We finally
derive an effective Brownian motion model from first principles which gives
good agreement with the simulations.Comment: accepted for publication in Eur. Phys. J.
Error-Correcting Codes and Minkowski’s Conjecture
The goal of this paper is twofold. The main one is to survey the latest results on the perfect and quasi-perfect Lee error correcting codes. The other goal is to show that the area of Lee error correcting codes, like many ideas in mathematics, can trace its roots to the Phytagorean theorem a2+b2 = c2. Thus to show that the area of the perfect Lee error correcting codes is an integral part of mathematics. It turns out that Minkowski’s conjecture, which is an interface of number theory, approximation theory, geometry, linear algebra, and group theory is one of the milestones on the route to Lee codes
Identifying and managing asbestiform minerals in geological collections
Asbestos is widely recognised as a serious hazard, and its industrial use is now banned within the UK, and EU, and strict regulations govern the use of older manufactured materials which may contain asbestos. However, asbestos is also a natural geological material, and may occur in museum collections as minerals or constituents of rock specimens. In the UK the Control of Asbestos Regulations 2012 (CAR 2012) provides the legal framework for the safe identification, use and disposal of asbestos. However, these regulations, and other EU regulations, provide no specific guidance on dealing with potentially asbestos-containing natural materials. CAR 2012 specifies just six asbestos minerals although a number of other minerals in museum collections are known to have asbestiform structures and be hazard-ous, including other amphiboles, and the zeolite erionite. Despite the lack of specific guid-ance, museums must comply with CAR 2012, and this paper outlines the professional ad-vice, training and procedures which may be needed for this. It provides guidance on identifi-cation of potential asbestos-bearing specimens and on procedures to document them and store them for future use, or to prepare them for professional disposal. It also makes sug-gestions how visitors, employees and others in a museum can be protected from asbestos as incoming donations and enquiries, managed in the event of an emergency, and safely included in displays
Fiber cavities with integrated mode matching optics
In fiber based Fabry-P\'{e}rot Cavities (FFPCs), limited spatial mode
matching between the cavity mode and input/output modes has been the main
hindrance for many applications. We have demonstrated a versatile mode matching
method for FFPCs. Our novel design employs an assembly of a graded-index and
large core multimode fiber directly spliced to a single mode fiber. This
all-fiber assembly transforms the propagating mode of the single mode fiber to
match with the mode of a FFPC. As a result, we have measured a mode matching of
90\% for a cavity length of 400 . This is a significant
improvement compared to conventional FFPCs coupled with just a single mode
fiber, especially at long cavity lengths. Adjusting the parameters of the
assembly, the fundamental cavity mode can be matched with the mode of almost
any single mode fiber, making this approach highly versatile and integrable.Comment: 6 pages, 5 figures, articl
- …