109 research outputs found
Theoretical Bounds and Constructions of Codes in the Generalized Cayley Metric
Permutation codes have recently garnered substantial research interest due to
their potential in various applications including cloud storage systems, genome
resequencing and flash memories. In this paper, we study the theoretical bounds
and constructions of permutation codes in the generalized Cayley metric. The
generalized Cayley metric captures the number of generalized transposition
errors in a permutation, and subsumes previously studied error types, including
transpositions and translocations, without imposing restrictions on the lengths
and positions of the translocated segments. Relying on the breakpoint analysis
proposed by Chee and Vu, we first propose a coding scheme that is order-optimal
albeit not constructive based on this method. We then develop another
construction of permutation codes in the generalized Cayley distance. This
scheme is both explicit and systematic. We also prove the existence of
order-optimal systematic codes and offer a concrete construction based on this
method. For the generalized Cayley metric, we prove that our coding schemes
have less redundancy than the existing codes based on interleaving when the
codelength is sufficiently large and the number of errors is relatively small.Comment: 18 pages; 0 figures; published on IEEE Transactions on Information
Theor
Multi-Dimensional Spatially-Coupled Code Design Through Informed Relocation of Circulants
A circulant-based spatially-coupled (SC) code is constructed by partitioning
the circulants of an underlying block code into a number of components, and
then coupling copies of these components together. By connecting (coupling)
several SC codes, multi-dimensional SC (MD-SC) codes are constructed. In this
paper, we present a systematic framework for constructing MD-SC codes with
notably better girth properties than their 1D-SC counterparts. In our
framework, informed multi-dimensional coupling is performed via an optimal
relocation and an (optional) power adjustment of problematic circulants in the
constituent SC codes. Compared to the 1D-SC codes, our MD-SC codes are
demonstrated to have up to 85% reduction in the population of the smallest
cycle, and up to 3.8 orders of magnitude BER improvement in the early error
floor region. The results of this work can be particularly beneficial in data
storage systems, e.g., 2D magnetic recording and 3D Flash systems, as
high-performance MD-SC codes are robust against various channel impairments and
non-uniformity.Comment: 7 pages, 9 figures, Allerton Conference 201
High Performance Non-Binary Spatially-Coupled Codes for Flash Memories
Modern dense Flash memory devices operate at very low error rates, which
require powerful error correcting coding (ECC) techniques. An emerging class of
graph-based ECC techniques that has broad applications is the class of
spatially-coupled (SC) codes, where a block code is partitioned into components
that are then rewired multiple times to construct an SC code. Here, our focus
is on SC codes with the underlying circulant-based structure. In this paper, we
present a three-stage approach for the design of high performance non-binary SC
(NB-SC) codes optimized for practical Flash channels; we aim at minimizing the
number of detrimental general absorbing sets of type two (GASTs) in the graph
of the designed NB-SC code. In the first stage, we deploy a novel partitioning
mechanism, called the optimal overlap partitioning, which acts on the
protograph of the SC code to produce optimal partitioning corresponding to the
smallest number of detrimental objects. In the second stage, we apply a new
circulant power optimizer to further reduce the number of detrimental GASTs. In
the third stage, we use the weight consistency matrix framework to manipulate
edge weights to eliminate as many as possible of the GASTs that remain in the
NB-SC code after the first two stages (that operate on the unlabeled graph of
the code). Simulation results reveal that NB-SC codes designed using our
approach outperform state-of-the-art NB-SC codes when used over Flash channels.Comment: 8 pages (double column), 5 figures, the short version was accepted at
the IEEE Information Theory Worksho
Finite-Length Construction of High Performance Spatially-Coupled Codes via Optimized Partitioning and Lifting
Spatially-coupled (SC) codes are a family of graph-based codes that have
attracted significant attention thanks to their capacity approaching
performance and low decoding latency. An SC code is constructed by partitioning
an underlying block code into a number of components and coupling their copies
together. In this paper, we first introduce a general approach for the
enumeration of detrimental combinatorial objects in the graph of finite-length
SC codes. Our approach is general in the sense that it effectively works for SC
codes with various column weights and memories. Next, we present a two-stage
framework for the construction of high-performance binary SC codes optimized
for additive white Gaussian noise channel; we aim at minimizing the number of
detrimental combinatorial objects in the error floor regime. In the first
stage, we deploy a novel partitioning scheme, called the optimal overlap
partitioning, to produce optimal partitioning corresponding to the smallest
number of detrimental objects. In the second stage, we apply a new circulant
power optimizer to further reduce the number of detrimental objects in the
lifted graph. An SC code constructed by our new framework has nearly 5 orders
of magnitudes error floor performance improvement compared to the uncoupled
setting.Comment: 30 pages; this manuscript is submitted to IEEE Transactions on
Communications (TCOM
A Deterministic Polynomial-Time Protocol for Synchronizing from Deletions
In this paper, we consider a synchronization problem between nodes and
that are connected through a two--way communication channel. {Node }
contains a binary file of length and {node } contains a binary file
that is generated by randomly deleting bits from , by a small deletion
rate . The location of deleted bits is not known to either node or
node . We offer a deterministic synchronization scheme between nodes and
that needs a total of transmitted bits and
reconstructs at node with probability of error that is exponentially
low in the size of . Orderwise, the rate of our scheme matches the optimal
rate for this channel.Comment: Accepted to the IEEE Transactions on Information Theor
Exact Reconstruction from Insertions in Synchronization Codes
This work studies problems in data reconstruction, an important area with
numerous applications. In particular, we examine the reconstruction of binary
and non-binary sequences from synchronization (insertion/deletion-correcting)
codes. These sequences have been corrupted by a fixed number of symbol
insertions (larger than the minimum edit distance of the code), yielding a
number of distinct traces to be used for reconstruction. We wish to know the
minimum number of traces needed for exact reconstruction. This is a general
version of a problem tackled by Levenshtein for uncoded sequences.
We introduce an exact formula for the maximum number of common supersequences
shared by sequences at a certain edit distance, yielding an upper bound on the
number of distinct traces necessary to guarantee exact reconstruction. Without
specific knowledge of the codewords, this upper bound is tight. We apply our
results to the famous single deletion/insertion-correcting Varshamov-Tenengolts
(VT) codes and show that a significant number of VT codeword pairs achieve the
worst-case number of outputs needed for exact reconstruction. We also consider
extensions to other channels, such as adversarial deletion and
insertion/deletion channels and probabilistic channels.Comment: 18 pages, 3 figures. Accepted to IEEE Transactions on Information
Theor
The Cycle Consistency Matrix Approach to Absorbing Sets in Separable Circulant-Based LDPC Codes
For LDPC codes operating over additive white Gaussian noise channels and
decoded using message-passing decoders with limited precision, absorbing sets
have been shown to be a key factor in error floor behavior. Focusing on this
scenario, this paper introduces the cycle consistency matrix (CCM) as a
powerful analytical tool for characterizing and avoiding absorbing sets in
separable circulant-based (SCB) LDPC codes. SCB codes include a wide variety of
regular LDPC codes such as array-based LDPC codes as well as many common
quasi-cyclic codes. As a consequence of its cycle structure, each potential
absorbing set in an SCB LDPC code has a CCM, and an absorbing set can be
present in an SCB LDPC code only if the associated CCM has a nontrivial null
space.
CCM-based analysis can determine the multiplicity of an absorbing set in an
SCB code and CCM-based constructions avoid certain small absorbing sets
completely. While these techniques can be applied to an SCB code of any rate,
lower-rate SCB codes can usually avoid small absorbing sets because of their
higher variable node degree. This paper focuses attention on the high-rate
scenario in which the CCM constructions provide the most benefit. Simulation
results demonstrate that under limited-precision decoding the new codes have
steeper error-floor slopes and can provide one order of magnitude of
improvement in the low FER region
Variability-Aware Read and Write Channel Models for 1S1R Crossbar Resistive Memory with High Wordline/Bitline Resistance
Crossbar resistive memory with 1 Selector 1 Resistor (1S1R) structure is
attractive for low-cost and high-density nonvolatile memory applications. As
technology scales down to the single-nm regime, the increasing resistivity of
wordline/bitline becomes a limiting factor to device reliability. This paper
presents write/read communication channels while considering the line
resistance and device variabilities by statistically relating the degraded
write/read margins and the channel parameters. Binary asymmetric channel (BAC)
models are proposed for the write/read operations and array capacity results
are presented. Simulations based on these models suggest that the bit-error
rate of devices are highly non-uniform across the memory array. These models
provide quantitative tools for evaluating the trade-offs between memory
reliability and design parameters, such as array size, technology nodes, and
aspect ratio, and also for designing coding-theoretic solutions that would be
most effective for crossbar memory
Coding for Deletion Channels with Multiple Traces
Motivated by the sequence reconstruction problem from traces in DNA-based
storage, we consider the problem of designing codes for the deletion channel
when multiple observations (or traces) are available to the decoder. We propose
simple binary and non-binary codes based on Varshamov-Tenengolts (VT) codes.
The proposed codes split the codeword in blocks and employ a VT code in each
block. The availability of multiple traces helps the decoder to identify
deletion-free copies of a block, and to avoid mis-synchronization while
decoding. The encoding complexity of the proposed scheme is linear in the
codeword length; the decoding complexity is linear in the codeword length, and
quadratic in the number of deletions and the number of traces. The proposed
scheme offers an explicit low-complexity technique for correcting deletions
using multiple traces.Comment: This paper will be presented at ISIT 201
Coding for Channels with SNR Variation: Spatial Coupling and Efficient Interleaving
In magnetic-recording systems, consecutive sections experience different
signal to noise ratios (SNRs). To perform error correction over these systems,
one approach is to use an individual block code for each section. However, the
performance over a section affected by a lower SNR is weaker compared to the
performance over a section affected by a higher SNR. Spatially-coupled (SC)
codes are a family of graph-based codes with capacity approaching performance
and low latency decoding. An SC code is constructed by partitioning an
underlying block code to several component matrices, and coupling copies of the
component matrices together. The contribution of this paper is threefold.
First, we present a new partitioning technique to efficiently construct SC
codes with column weights 4 and 6. Second, we present an SC code construction
for channels with SNR variation. Our SC code construction provides local error
correction for each section by means of the underlying codes that cover one
section each, and simultaneously, an added level of error correction by means
of coupling among the underlying codes. Third, we introduce a low-complexity
interleaving scheme specific to SC codes that further improves their
performance over channels with SNR variation. Our simulation results show that
our SC codes outperform individual block codes by more than 1 and 2 orders of
magnitudes in the error floor region compared to the block codes with and
without regular interleaving, respectively. This improvement is more pronounced
by increasing the memory and column weight.Comment: 8 pages, Submitted to IEEE Transactions on Magnetics (TMAG
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