248 research outputs found
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
Minimum Pseudo-Weight and Minimum Pseudo-Codewords of LDPC Codes
In this correspondence, we study the minimum pseudo-weight and minimum
pseudo-codewords of low-density parity-check (LDPC) codes under linear
programming (LP) decoding. First, we show that the lower bound of Kelly,
Sridhara, Xu and Rosenthal on the pseudo-weight of a pseudo-codeword of an LDPC
code with girth greater than 4 is tight if and only if this pseudo-codeword is
a real multiple of a codeword. Then, we show that the lower bound of Kashyap
and Vardy on the stopping distance of an LDPC code is also a lower bound on the
pseudo-weight of a pseudo-codeword of this LDPC code with girth 4, and this
lower bound is tight if and only if this pseudo-codeword is a real multiple of
a codeword. Using these results we further show that for some LDPC codes, there
are no other minimum pseudo-codewords except the real multiples of minimum
codewords. This means that the LP decoding for these LDPC codes is
asymptotically optimal in the sense that the ratio of the probabilities of
decoding errors of LP decoding and maximum-likelihood decoding approaches to 1
as the signal-to-noise ratio leads to infinity. Finally, some LDPC codes are
listed to illustrate these results.Comment: 17 pages, 1 figur
Pseudo-Codeword Analysis of Tanner Graphs from Projective and Euclidean Planes
In order to understand the performance of a code under maximum-likelihood
(ML) decoding, one studies the codewords, in particular the minimal codewords,
and their Hamming weights. In the context of linear programming (LP) decoding,
one's attention needs to be shifted to the pseudo-codewords, in particular to
the minimal pseudo-codewords, and their pseudo-weights. In this paper we
investigate some families of codes that have good properties under LP decoding,
namely certain families of low-density parity-check (LDPC) codes that are
derived from projective and Euclidean planes: we study the structure of their
minimal pseudo-codewords and give lower bounds on their pseudo-weight.Comment: Submitted to IEEE Transactions on Information Theory, February 25,
200
On the Pseudocodeword Redundancy of Binary Linear Codes
The AWGNC, BSC, and max-fractional pseudocodeword redundancies of a binary
linear code are defined to be the smallest number of rows in a parity-check
matrix such that the corresponding minimum pseudoweight is equal to the minimum
Hamming distance of the code. It is shown that most codes do not have a finite
pseudocodeword redundancy. Also, upper bounds on the pseudocodeword redundancy
for some families of codes, including codes based on designs, are provided. The
pseudocodeword redundancies for all codes of small length (at most 9) are
computed. Furthermore, comprehensive results are provided on the cases of
cyclic codes of length at most 250 for which the eigenvalue bound of Vontobel
and Koetter is sharp.Comment: 14 pages. arXiv admin note: substantial text overlap with
arXiv:1005.348
Provably efficient instanton search algorithm for LP decoding of LDPC codes over the BSC
We consider Linear Programming (LP) decoding of a fixed Low-Density
Parity-Check (LDPC) code over the Binary Symmetric Channel (BSC). The LP
decoder fails when it outputs a pseudo-codeword which is not a codeword. We
design an efficient algorithm termed the Instanton Search Algorithm (ISA)
which, given a random input, generates a set of flips called the BSC-instanton.
We prove that: (a) the LP decoder fails for any set of flips with support
vector including an instanton; (b) for any input, the algorithm outputs an
instanton in the number of steps upper-bounded by twice the number of flips in
the input. Repeated sufficient number of times, the ISA outcomes the number of
unique instantons of different sizes.Comment: Submitted to IEEE Transactions on Information Theory. 9 Pages, 4
Figures; Dr. Bane Vasic added as an author; Changes made to the introduction
and abstract; Acknowledgment section added; Some references added; Figures
modified to make them more clear
Exploration of AWGNC and BSC Pseudocodeword Redundancy
The AWGNC, BSC, and max-fractional pseudocodeword redundancy of a code is
defined as the smallest number of rows in a parity-check matrix such that the
corresponding minimum pseudoweight is equal to the minimum Hamming distance of
the code. This paper provides new results on the AWGNC, BSC, and max-fractional
pseudocodeword redundancies of codes. The pseudocodeword redundancies for all
codes of small length (at most 9) are computed. Also, comprehensive results are
provided on the cases of cyclic codes of length at most 250 for which the
eigenvalue bound of Vontobel and Koetter is sharp.Comment: 7 page
Even-freeness of cyclic 2-designs
A Steiner 2-design of block size k is an ordered pair (V, B) of finite sets
such that B is a family of k-subsets of V in which each pair of elements of V
appears exactly once. A Steiner 2-design is said to be r-even-free if for every
positive integer i =< r it contains no set of i elements of B in which each
element of V appears exactly even times. We study the even-freeness of a
Steiner 2-design when the cyclic group acts regularly on V. We prove the
existence of infinitely many nontrivial Steiner 2-designs of large block size
which have the cyclic automorphisms and higher even-freeness than the trivial
lower bound but are not the points and lines of projective geometry.Comment: 12 pages, no figure
On Quantum and Classical Error Control Codes: Constructions and Applications
It is conjectured that quantum computers are able to solve certain problems
more quickly than any deterministic or probabilistic computer. A quantum
computer exploits the rules of quantum mechanics to speed up computations.
However, it is a formidable task to build a quantum computer, since the quantum
mechanical systems storing the information unavoidably interact with their
environment. Therefore, one has to mitigate the resulting noise and decoherence
effects to avoid computational errors.
In this work, I study various aspects of quantum error control codes -- the
key component of fault-tolerant quantum information processing. I present the
fundamental theory and necessary background of quantum codes and construct many
families of quantum block and convolutional codes over finite fields, in
addition to families of subsystem codes over symmetric and asymmetric channels.
Particularly, many families of quantum BCH, RS, duadic, and convolutional
codes are constructed over finite fields. Families of subsystem codes and a
class of optimal MDS subsystem codes are derived over asymmetric and symmetric
quantum channels. In addition, propagation rules and tables of upper bounds on
subsystem code parameters are established. Classes of quantum and classical
LDPC codes based on finite geometries and Latin squares are constructed.Comment: Parts of PhD dissertation, Texas A&M Universit
Coding Theory and Projective Spaces
The projective space of order over a finite field \F_q is a set of all
subspaces of the vector space \F_q^{n}. In this work, we consider
error-correcting codes in the projective space, focusing mainly on constant
dimension codes. We start with the different representations of subspaces in
the projective space. These representations involve matrices in reduced row
echelon form, associated binary vectors, and Ferrers diagrams. Based on these
representations, we provide a new formula for the computation of the distance
between any two subspaces in the projective space. We examine lifted maximum
rank distance (MRD) codes, which are nearly optimal constant dimension codes.
We prove that a lifted MRD code can be represented in such a way that it forms
a block design known as a transversal design. The incidence matrix of the
transversal design derived from a lifted MRD code can be viewed as a
parity-check matrix of a linear code in the Hamming space. We find the
properties of these codes which can be viewed also as LDPC codes. We present
new bounds and constructions for constant dimension codes. First, we present a
multilevel construction for constant dimension codes, which can be viewed as a
generalization of a lifted MRD codes construction. This construction is based
on a new type of rank-metric codes, called Ferrers diagram rank-metric codes.
Then we derive upper bounds on the size of constant dimension codes which
contain the lifted MRD code, and provide a construction for two families of
codes, that attain these upper bounds. We generalize the well-known concept of
a punctured code for a code in the projective space to obtain large codes which
are not constant dimension. We present efficient enumerative encoding and
decoding techniques for the Grassmannian. Finally we describe a search method
for constant dimension lexicodes.Comment: This is a PhD thesis performed at the Technion by Natalia Silberstein
and supervised by Prof. Tuvi Etzio
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