3,893 research outputs found
Refined Upper Bounds on Stopping Redundancy of Binary Linear Codes
The -th stopping redundancy of the binary
code , , is defined as the minimum number of rows in
the parity-check matrix of , such that the smallest stopping set is
of size at least . The stopping redundancy is defined as
. In this work, we improve on the probabilistic analysis of
stopping redundancy, proposed by Han, Siegel and Vardy, which yields the best
bounds known today. In our approach, we judiciously select the first few rows
in the parity-check matrix, and then continue with the probabilistic method. By
using similar techniques, we improve also on the best known bounds on
, for . Our approach is compared to the
existing methods by numerical computations.Comment: 5 pages; ITW 201
The Trapping Redundancy of Linear Block Codes
We generalize the notion of the stopping redundancy in order to study the
smallest size of a trapping set in Tanner graphs of linear block codes. In this
context, we introduce the notion of the trapping redundancy of a code, which
quantifies the relationship between the number of redundant rows in any
parity-check matrix of a given code and the size of its smallest trapping set.
Trapping sets with certain parameter sizes are known to cause error-floors in
the performance curves of iterative belief propagation decoders, and it is
therefore important to identify decoding matrices that avoid such sets. Bounds
on the trapping redundancy are obtained using probabilistic and constructive
methods, and the analysis covers both general and elementary trapping sets.
Numerical values for these bounds are computed for the [2640,1320] Margulis
code and the class of projective geometry codes, and compared with some new
code-specific trapping set size estimates.Comment: 12 pages, 4 tables, 1 figure, accepted for publication in IEEE
Transactions on Information Theor
Permutation Decoding and the Stopping Redundancy Hierarchy of Cyclic and Extended Cyclic Codes
We introduce the notion of the stopping redundancy hierarchy of a linear
block code as a measure of the trade-off between performance and complexity of
iterative decoding for the binary erasure channel. We derive lower and upper
bounds for the stopping redundancy hierarchy via Lovasz's Local Lemma and
Bonferroni-type inequalities, and specialize them for codes with cyclic
parity-check matrices. Based on the observed properties of parity-check
matrices with good stopping redundancy characteristics, we develop a novel
decoding technique, termed automorphism group decoding, that combines iterative
message passing and permutation decoding. We also present bounds on the
smallest number of permutations of an automorphism group decoder needed to
correct any set of erasures up to a prescribed size. Simulation results
demonstrate that for a large number of algebraic codes, the performance of the
new decoding method is close to that of maximum likelihood decoding.Comment: 40 pages, 6 figures, 10 tables, submitted to IEEE Transactions on
Information Theor
Optimization of Parity-Check Matrices of LDPC Codes
Madala tihedusega paarsuskontroll (LDPC) on laialdaselt kasutusel kommunikatsioonis
tänu oma suurepärasele praktilisele võimekusele. LDPC koodi vigade tõenäosust iteratiivse
dekodeerimise puhul binaarsel kustutuskanalil määrab klass kombinatoorseid objekte, nimega
peatamise rühm. Väikese suurusega peatamise rühmad on dekodeerija vigade põhjuseks.
Peatamise liiasust määratletakse kui minimaalset ridade arvu paarsuskontrolli koodi
maatriksis, mille puhul pole selles väikesi peatuse rühmi.
Han, Siegel ja Vardy kasutavad üld binaarse lineaarkoodi ülemise piiri peatamiste liiasuse
tuletamiseks tõenäosuslikku analüüsi. Need piirid on teadaolevalt parimad paljude koodi
perekondade puhul. Selles töös me parendame Hani, Siegeli ja Vardy tulemusi modifitseerides
selleks nende analüüsi. Meie lähenemine erineb sellepoolest, et me valime mõistlikult esimese
ja teise rea paarsuskontrolli maatriksis ja siis läheme edasi tõenäosusliku analüüsiga.
Numbrilised väärtused kinnitavad seda, et piirid mis on määratletud selles töös on paremad
Hani, Siegeli ja Vardy omadest kahe koodi puhul: laiendatud Golay koodis ja kvadraatses jääk
koodis pikkusega 48.Low-density parity-check (LDPC) codes are widely used in communications due to their excellent practical performance. Error probability of LDPC code under iterative decoding on the binary erasure channel is determined by a class of combinatorial objects, called stopping sets. Stopping sets of small size are the reason for the decoder failures. Stopping redundancy is defined as the minimum number of rows in a parity-check matrix of the code, such that there are no small stopping sets in it.
Han, Siegel and Vardy derive upper bounds on the stopping redundancy of general binary linear codes by using probabilistic analysis. For many families of codes, these bounds are the best currently known. In this work, we improve on the results of Han, Siegel and Vardy by modifying their analysis. Our approach is different in that we judiciously select the first and the second rows in the parity-check matrix, and then proceed with the probabilistic analysis. Numerical experiments confirm that the bounds obtained in this thesis are superior to those of Han, Siegel and Vardy for two codes: the extended Golay code and the quadratic residue code of length 48
Stopping Set Distributions of Some Linear Codes
Stopping sets and stopping set distribution of an low-density parity-check
code are used to determine the performance of this code under iterative
decoding over a binary erasure channel (BEC). Let be a binary
linear code with parity-check matrix , where the rows of may be
dependent. A stopping set of with parity-check matrix is a subset
of column indices of such that the restriction of to does not
contain a row of weight one. The stopping set distribution
enumerates the number of stopping sets with size of with parity-check
matrix . Note that stopping sets and stopping set distribution are related
to the parity-check matrix of . Let be the parity-check matrix
of which is formed by all the non-zero codewords of its dual code
. A parity-check matrix is called BEC-optimal if
and has the smallest number of rows. On the
BEC, iterative decoder of with BEC-optimal parity-check matrix is an
optimal decoder with much lower decoding complexity than the exhaustive
decoder. In this paper, we study stopping sets, stopping set distributions and
BEC-optimal parity-check matrices of binary linear codes. Using finite geometry
in combinatorics, we obtain BEC-optimal parity-check matrices and then
determine the stopping set distributions for the Simplex codes, the Hamming
codes, the first order Reed-Muller codes and the extended Hamming codes.Comment: 33 pages, submitted to IEEE Trans. Inform. Theory, Feb. 201
On generic erasure correcting sets and related problems
Motivated by iterative decoding techniques for the binary erasure channel
Hollmann and Tolhuizen introduced and studied the notion of generic erasure
correcting sets for linear codes. A generic --erasure correcting set
generates for all codes of codimension a parity check matrix that allows
iterative decoding of all correctable erasure patterns of size or less. The
problem is to derive bounds on the minimum size of generic erasure
correcting sets and to find constructions for such sets. In this paper we
continue the study of these sets. We derive better lower and upper bounds.
Hollmann and Tolhuizen also introduced the stronger notion of --sets and
derived bounds for their minimum size . Here also we improve these
bounds. We observe that these two conceps are closely related to so called
--wise intersecting codes, an area, in which has been studied
primarily with respect to ratewise performance. We derive connections. Finally,
we observed that hypergraph covering can be used for both problems to derive
good upper bounds.Comment: 9 pages, to appear in IEEE Transactions on Information Theor
Invertible Bloom Lookup Tables with Listing Guarantees
The Invertible Bloom Lookup Table (IBLT) is a probabilistic concise data
structure for set representation that supports a listing operation as the
recovery of the elements in the represented set. Its applications can be found
in network synchronization and traffic monitoring as well as in
error-correction codes. IBLT can list its elements with probability affected by
the size of the allocated memory and the size of the represented set, such that
it can fail with small probability even for relatively small sets. While
previous works only studied the failure probability of IBLT, this work
initiates the worst case analysis of IBLT that guarantees successful listing
for all sets of a certain size. The worst case study is important since the
failure of IBLT imposes high overhead. We describe a novel approach that
guarantees successful listing when the set satisfies a tunable upper bound on
its size. To allow that, we develop multiple constructions that are based on
various coding techniques such as stopping sets and the stopping redundancy of
error-correcting codes, Steiner systems, and covering arrays as well as new
methodologies we develop. We analyze the sizes of IBLTs with listing guarantees
obtained by the various methods as well as their mapping memory consumption.
Lastly, we study lower bounds on the achievable sizes of IBLT with listing
guarantees and verify the results in the paper by simulations
Density Evolution and Functional Threshold for the Noisy Min-Sum Decoder
This paper investigates the behavior of the Min-Sum decoder running on noisy
devices. The aim is to evaluate the robustness of the decoder in the presence
of computation noise, e.g. due to faulty logic in the processing units, which
represents a new source of errors that may occur during the decoding process.
To this end, we first introduce probabilistic models for the arithmetic and
logic units of the the finite-precision Min-Sum decoder, and then carry out the
density evolution analysis of the noisy Min-Sum decoder. We show that in some
particular cases, the noise introduced by the device can help the Min-Sum
decoder to escape from fixed points attractors, and may actually result in an
increased correction capacity with respect to the noiseless decoder. We also
reveal the existence of a specific threshold phenomenon, referred to as
functional threshold. The behavior of the noisy decoder is demonstrated in the
asymptotic limit of the code-length -- by using "noisy" density evolution
equations -- and it is also verified in the finite-length case by Monte-Carlo
simulation.Comment: 46 pages (draft version); extended version of the paper with same
title, submitted to IEEE Transactions on Communication
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