414 research outputs found
Constant-Weight Gray Codes for Local Rank Modulation
We consider the local rank-modulation scheme in which a sliding window going
over a sequence of real-valued variables induces a sequence of permutations.
The local rank-modulation, as a generalization of the rank-modulation scheme,
has been recently suggested as a way of storing information in flash memory.
We study constant-weight Gray codes for the local rank-modulation scheme in
order to simulate conventional multi-level flash cells while retaining the
benefits of rank modulation. We provide necessary conditions for the existence
of cyclic and cyclic optimal Gray codes. We then specifically study codes of
weight 2 and upper bound their efficiency, thus proving that there are no such
asymptotically-optimal cyclic codes. In contrast, we study codes of weight 3
and efficiently construct codes which are asymptotically-optimal
Van der Waals Interaction between Flux Lines in High-T_c Superconductors: A Variational Approach
In pure anisotropic or layered superconductors thermal fluctuations induce a
van der Waals attraction between flux lines. This attraction together with the
entropic repulsion has interesting consequences for the low field phase
diagram; in particular, a first order transition from the Meissner phase to the
mixed state is induced. We introduce a new variational approach that allows for
the calculation of the effective free energy of the flux line lattice on the
scale of the mean flux line distance, which is based on an expansion of the
free energy around the regular triangular Abrikosov lattice. Using this
technique, the low field phase diagram of these materials may be explored. The
results of this technique are compared with a recent functional RG treatment of
the same system.Comment: 8 pages, 7 figure
On Optimal Anticodes over Permutations with the Infinity Norm
Motivated by the set-antiset method for codes over permutations under the
infinity norm, we study anticodes under this metric. For half of the parameter
range we classify all the optimal anticodes, which is equivalent to finding the
maximum permanent of certain -matrices. For the rest of the cases we
show constraints on the structure of optimal anticodes
Limited-Magnitude Error-Correcting Gray Codes for Rank Modulation
We construct Gray codes over permutations for the rank-modulation scheme,
which are also capable of correcting errors under the infinity-metric. These
errors model limited-magnitude or spike errors, for which only
single-error-detecting Gray codes are currently known. Surprisingly, the
error-correcting codes we construct achieve a better asymptotic rate than that
of presently known constructions not having the Gray property, and exceed the
Gilbert-Varshamov bound. Additionally, we present efficient ranking and
unranking procedures, as well as a decoding procedure that runs in linear time.
Finally, we also apply our methods to solve an outstanding issue with
error-detecting rank-modulation Gray codes (snake-in-the-box codes) under a
different metric, the Kendall -metric, in the group of permutations over
an even number of elements , where we provide asymptotically optimal
codes.Comment: Revised version for journal submission. Additional results include
more tight auxiliary constructions, a decoding shcema, ranking/unranking
procedures, and application to snake-in-the-box codes under the Kendall
tau-metri
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