473 research outputs found
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
Trellis coded 4-ary PAM using distance preserving mapping
Abstract: A trellis coded 4-ary Pulse Amplitude Modulation (4-PAM) is presented, where the encoding algorithm is derived from Distance Preserving Mapping (DPM) algorithm. In this work, we modify the DPM algorithm for 4-PAM and obtain a new construction for mapping binary sequences to permutation sequences, where the permutation sequences are obtained by permuting symbols of a 4-PAM constellation. The resulting codebook of permutation sequences formed this way are termed mappings. We also present several metrics for assessing the performance of the mappings from our construction, and we show that a metric called the Sum of Product of Distances (SOPD) is the best metric to use when judging the performance of the mappings. Finally, performance results are presented, where the mappings from our construction are compared against each other and also against the conventional mappings in the literature. Index Terms—Distance-preserving mappings, Hamming distance, Euclidean distance, Pulse Amplitude Modulation
On the Combinatorial Version of the Slepian-Wolf Problem
We study the following combinatorial version of the Slepian-Wolf coding
scheme. Two isolated Senders are given binary strings and respectively;
the length of each string is equal to , and the Hamming distance between the
strings is at most . The Senders compress their strings and
communicate the results to the Receiver. Then the Receiver must reconstruct
both strings and . The aim is to minimise the lengths of the transmitted
messages.
For an asymmetric variant of this problem (where one of the Senders transmits
the input string to the Receiver without compression) with deterministic
encoding a nontrivial lower bound was found by A.Orlitsky and K.Viswanathany.
In our paper we prove a new lower bound for the schemes with syndrome coding,
where at least one of the Senders uses linear encoding of the input string.
For the combinatorial Slepian-Wolf problem with randomized encoding the
theoretical optimum of communication complexity was recently found by the first
author, though effective protocols with optimal lengths of messages remained
unknown. We close this gap and present a polynomial time randomized protocol
that achieves the optimal communication complexity.Comment: 20 pages, 14 figures. Accepted to IEEE Transactions on Information
Theory (June 2018
Analysis of permutation distance-preserving mappings using graphs
Abstract A new way of analyzing permutation distance preserving mappings is presented by making use of a graph representation. The properties necessary to make such graphs distance-preserving and how this relates to the total sum of distances that exist for such mappings, are investigated. This new knowledge is used to analyze previous constructions, as well as showing the existence or non-existence of simple algorithms for mappings attaining the upper bound on the sum of distances. Finally, two applications for such graphs are considered
Estimates on the Size of Symbol Weight Codes
The study of codes for powerlines communication has garnered much interest
over the past decade. Various types of codes such as permutation codes,
frequency permutation arrays, and constant composition codes have been proposed
over the years. In this work we study a type of code called the bounded symbol
weight codes which was first introduced by Versfeld et al. in 2005, and a
related family of codes that we term constant symbol weight codes. We provide
new upper and lower bounds on the size of bounded symbol weight and constant
symbol weight codes. We also give direct and recursive constructions of codes
for certain parameters.Comment: 14 pages, 4 figure
New distance concept and graph theory approach for certain coding techniques design and analysis
Abstract: A New graph distance concept introduced for certain coding techniques helped in their design and analysis as in the case of distance-preserving mappings and spectral shaping codes. A graph theoretic construction, mapping binary sequences to permutation sequences and inspired from the k-cube graph has reached the upper bound on the sum of the distances for certain values of the length of the permutation sequence. The new introduced distance concept in the k-cube graph helped better understanding and analyzing for the first time the concept of distance-reducing mappings. A combination of distance and the index-permutation graph concepts helped uncover and verify certain properties of spectral null codes, which were previously difficult to analyze
Using graphs for the analysis and construction of permutation distance-preserving mappings
Abstract: A new way of looking at permutation distance-preserving mappings (DPMs) is presented by making use of a graph representation. The properties necessary to make such a graph distance-preserving, are also investigated. Further, this new knowledge is used to analyze previous constructions, as well as to construct a new general mapping algorithm for a previous multilevel construction
Combinatorial Methods in Coding Theory
This thesis is devoted to a range of questions in applied mathematics and signal processing motivated by applications in error correction, compressed sensing, and writing on non-volatile memories. The underlying thread of our results is the use of diverse combinatorial methods originating in coding theory and computer science.
The thesis addresses three groups of problems. The first of them is
aimed at the construction and analysis of codes for error correction. Here we examine properties of codes that are constructed using random and structured graphs and hypergraphs, with the main purpose of devising new decoding algorithms as well as estimating the distribution of Hamming weights in the resulting codes. Some of the results obtained give the best known estimates of the number of correctable errors for codes whose decoding relies on local operations on the graph.
In the second part we address the question of constructing sampling
operators for the compressed sensing problem. This topic has been
the subject of a large body of works in the literature. We propose
general constructions of sampling matrices based on ideas from coding theory that act as near-isometric maps on almost all sparse signal. This matrices can be used for dimensionality reduction and compressed sensing.
In the third part we study the problem of reliable storage of information in non-volatile memories such as flash drives. This problem gives rise to a writing scheme that relies on relative magnitudes of neighboring cells, known as rank modulation. We establish the exact asymptotic behavior of the size of codes for rank modulation and suggest a number of new general constructions of such codes based on properties of finite fields as well as combinatorial considerations
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