2,286 research outputs found
Constructions of Rank Modulation Codes
Rank modulation is a way of encoding information to correct errors in flash
memory devices as well as impulse noise in transmission lines. Modeling rank
modulation involves construction of packings of the space of permutations
equipped with the Kendall tau distance.
We present several general constructions of codes in permutations that cover
a broad range of code parameters. In particular, we show a number of ways in
which conventional error-correcting codes can be modified to correct errors in
the Kendall space. Codes that we construct afford simple encoding and decoding
algorithms of essentially the same complexity as required to correct errors in
the Hamming metric. For instance, from binary BCH codes we obtain codes
correcting Kendall errors in memory cells that support the order of
messages, for any constant We also construct
families of codes that correct a number of errors that grows with at
varying rates, from to . One of our constructions
gives rise to a family of rank modulation codes for which the trade-off between
the number of messages and the number of correctable Kendall errors approaches
the optimal scaling rate. Finally, we list a number of possibilities for
constructing codes of finite length, and give examples of rank modulation codes
with specific parameters.Comment: Submitted to IEEE Transactions on Information Theor
Systematic Codes for Rank Modulation
The goal of this paper is to construct systematic error-correcting codes for
permutations and multi-permutations in the Kendall's -metric. These codes
are important in new applications such as rank modulation for flash memories.
The construction is based on error-correcting codes for multi-permutations and
a partition of the set of permutations into error-correcting codes. For a given
large enough number of information symbols , and for any integer , we
present a construction for systematic -error-correcting codes,
for permutations from , with less redundancy symbols than the number
of redundancy symbols in the codes of the known constructions. In particular,
for a given and for sufficiently large we can obtain . The same
construction is also applied to obtain related systematic error-correcting
codes for multi-permutations.Comment: to be presented ISIT201
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
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
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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