163 research outputs found
New Bounds for Permutation Codes in Ulam Metric
New bounds on the cardinality of permutation codes equipped with the Ulam
distance are presented. First, an integer-programming upper bound is derived,
which improves on the Singleton-type upper bound in the literature for some
lengths. Second, several probabilistic lower bounds are developed, which
improve on the known lower bounds for large minimum distances. The results of a
computer search for permutation codes are also presented.Comment: To be presented at ISIT 2015, 5 page
Error-Correction in Flash Memories via Codes in the Ulam Metric
We consider rank modulation codes for flash memories that allow for handling
arbitrary charge-drop errors. Unlike classical rank modulation codes used for
correcting errors that manifest themselves as swaps of two adjacently ranked
elements, the proposed \emph{translocation rank codes} account for more general
forms of errors that arise in storage systems. Translocations represent a
natural extension of the notion of adjacent transpositions and as such may be
analyzed using related concepts in combinatorics and rank modulation coding.
Our results include derivation of the asymptotic capacity of translocation rank
codes, construction techniques for asymptotically good codes, as well as simple
decoding methods for one class of constructed codes. As part of our exposition,
we also highlight the close connections between the new code family and
permutations with short common subsequences, deletion and insertion
error-correcting codes for permutations, and permutation codes in the Hamming
distance
Systematic Error-Correcting Codes for Rank Modulation
The rank-modulation scheme has been recently proposed for efficiently storing
data in nonvolatile memories. Error-correcting codes are essential for rank
modulation, however, existing results have been limited. In this work we
explore a new approach, \emph{systematic error-correcting codes for rank
modulation}. Systematic codes have the benefits of enabling efficient
information retrieval and potentially supporting more efficient encoding and
decoding procedures. We study systematic codes for rank modulation under
Kendall's -metric as well as under the -metric.
In Kendall's -metric we present -systematic codes for
correcting one error, which have optimal rates, unless systematic perfect codes
exist. We also study the design of multi-error-correcting codes, and provide
two explicit constructions, one resulting in systematic codes
with redundancy at most . We use non-constructive arguments to show the
existence of -systematic codes for general parameters. Furthermore,
we prove that for rank modulation, systematic codes achieve the same capacity
as general error-correcting codes.
Finally, in the -metric we construct two systematic
multi-error-correcting codes, the first for the case of , and the
second for . In the latter case, the codes have the same
asymptotic rate as the best codes currently known in this metric
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