34 research outputs found
Design of LDPC codes and reliable practical decoders for standard and non-standard channels
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Coding Theory
Coding theory lies naturally at the intersection of a large number of disciplines in pure and applied mathematics: algebra and number theory, probability theory and statistics, communication theory, discrete mathematics and combinatorics, complexity theory, and statistical physics. The workshop on coding theory covered many facets of the recent research advances
A STUDY OF LINEAR ERROR CORRECTING CODES
Since Shannon's ground-breaking work in 1948, there have been two main development streams
of channel coding in approaching the limit of communication channels, namely classical coding
theory which aims at designing codes with large minimum Hamming distance and probabilistic
coding which places the emphasis on low complexity probabilistic decoding using long codes built
from simple constituent codes. This work presents some further investigations in these two channel
coding development streams.
Low-density parity-check (LDPC) codes form a class of capacity-approaching codes with sparse
parity-check matrix and low-complexity decoder Two novel methods of constructing algebraic binary
LDPC codes are presented. These methods are based on the theory of cyclotomic cosets, idempotents
and Mattson-Solomon polynomials, and are complementary to each other. The two methods
generate in addition to some new cyclic iteratively decodable codes, the well-known Euclidean and
projective geometry codes. Their extension to non binary fields is shown to be straightforward.
These algebraic cyclic LDPC codes, for short block lengths, converge considerably well under iterative
decoding. It is also shown that for some of these codes, maximum likelihood performance may
be achieved by a modified belief propagation decoder which uses a different subset of 7^ codewords
of the dual code for each iteration.
Following a property of the revolving-door combination generator, multi-threaded minimum
Hamming distance computation algorithms are developed. Using these algorithms, the previously
unknown, minimum Hamming distance of the quadratic residue code for prime 199 has been evaluated.
In addition, the highest minimum Hamming distance attainable by all binary cyclic codes
of odd lengths from 129 to 189 has been determined, and as many as 901 new binary linear codes
which have higher minimum Hamming distance than the previously considered best known linear
code have been found.
It is shown that by exploiting the structure of circulant matrices, the number of codewords
required, to compute the minimum Hamming distance and the number of codewords of a given
Hamming weight of binary double-circulant codes based on primes, may be reduced. A means
of independently verifying the exhaustively computed number of codewords of a given Hamming
weight of these double-circulant codes is developed and in coiyunction with this, it is proved that
some published results are incorrect and the correct weight spectra are presented. Moreover, it is
shown that it is possible to estimate the minimum Hamming distance of this family of prime-based
double-circulant codes.
It is shown that linear codes may be efficiently decoded using the incremental correlation Dorsch
algorithm. By extending this algorithm, a list decoder is derived and a novel, CRC-less error detection
mechanism that offers much better throughput and performance than the conventional ORG
scheme is described. Using the same method it is shown that the performance of conventional CRC
scheme may be considerably enhanced. Error detection is an integral part of an incremental redundancy
communications system and it is shown that sequences of good error correction codes,
suitable for use in incremental redundancy communications systems may be obtained using the
Constructions X and XX. Examples are given and their performances presented in comparison to
conventional CRC schemes
A Comparison Study of LDPC and BCH Codes
The need for efficient and reliable digital data communication systems has been rising
rapidly in recent years. There are various reasons that have brought this need for the
communication systems, among them are the increase in automatic data processing
equipment and the increased need for long range communication. Therefore, the
LDPC and BCH codes were developed for achieving more reliable data transmission
in communication systems. This project covers the research about the LDPC and
BCH error correction codes. Algorithm for simulating both the LDPC and BCH
codes were also being investigated, which includes generating the parity check
matrix, generating the message code in Galois array matrix, encoding the message
bits, modulation and decoding the message bits for LDPC. Matlab software is used
for encoding and decoding the codes. The percentage of accuracy for LDPC
simulation codes are ranging from 95% to 99%. The results obtained shows that the
LDPC codes are more efficient and reliable than the BCH codes coding method of
error correction because the LDPC codes had a channel performance very close to the
Shannon limit. LDPC codes are a class of linear block codes that are proving to be
the best performing forward error correction available. Markets such as broadband
wireless and mobile networks operate in noisy environments and need powerful error
correction in order to improve reliability and better data rates. Through LDPC and
BCH codes, these systems can operate more reliably, efficiently and at higher data
rates
A STUDY OF ERASURE CORRECTING CODES
This work focus on erasure codes, particularly those that of high performance,
and the related decoding algorithms, especially with low
computational complexity. The work is composed of different pieces,
but the main components are developed within the following two main
themes.
Ideas of message passing are applied to solve the erasures after the
transmission. Efficient matrix-representation of the belief propagation
(BP) decoding algorithm on the BEG is introduced as the recovery
algorithm. Gallager's bit-flipping algorithm are further developed
into the guess and multi-guess algorithms especially for the
application to recover the unsolved erasures after the recovery algorithm.
A novel maximum-likelihood decoding algorithm, the In-place
algorithm, is proposed with a reduced computational complexity. A
further study on the marginal number of correctable erasures by the
In-place algoritinn determines a lower bound of the average number
of correctable erasures. Following the spirit in search of the most likable
codeword based on the received vector, we propose a new branch-evaluation-
search-on-the-code-tree (BESOT) algorithm, which is powerful
enough to approach the ML performance for all linear block
codes.
To maximise the recovery capability of the In-place algorithm in
network transmissions, we propose the product packetisation structure
to reconcile the computational complexity of the In-place algorithm.
Combined with the proposed product packetisation structure,
the computational complexity is less than the quadratic complexity
bound. We then extend this to application of the Rayleigh fading
channel to solve the errors and erasures. By concatenating an outer
code, such as BCH codes, the product-packetised RS codes have the
performance of the hard-decision In-place algorithm significantly better
than that of the soft-decision iterative algorithms on optimally
designed LDPC codes
Some Notes on Code-Based Cryptography
This thesis presents new cryptanalytic results in several areas of coding-based cryptography. In addition, we also investigate the possibility of using convolutional codes in code-based public-key cryptography. The first algorithm that we present is an information-set decoding algorithm, aiming towards the problem of decoding random linear codes. We apply the generalized birthday technique to information-set decoding, improving the computational complexity over previous approaches. Next, we present a new version of the McEliece public-key cryptosystem based on convolutional codes. The original construction uses Goppa codes, which is an algebraic code family admitting a well-defined code structure. In the two constructions proposed, large parts of randomly generated parity checks are used. By increasing the entropy of the generator matrix, this presumably makes structured attacks more difficult. Following this, we analyze a McEliece variant based on quasi-cylic MDPC codes. We show that when the underlying code construction has an even dimension, the system is susceptible to, what we call, a squaring attack. Our results show that the new squaring attack allows for great complexity improvements over previous attacks on this particular McEliece construction. Then, we introduce two new techniques for finding low-weight polynomial multiples. Firstly, we propose a general technique based on a reduction to the minimum-distance problem in coding, which increases the multiplicity of the low-weight codeword by extending the code. We use this algorithm to break some of the instances used by the TCHo cryptosystem. Secondly, we propose an algorithm for finding weight-4 polynomials. By using the generalized birthday technique in conjunction with increasing the multiplicity of the low-weight polynomial multiple, we obtain a much better complexity than previously known algorithms. Lastly, two new algorithms for the learning parities with noise (LPN) problem are proposed. The first one is a general algorithm, applicable to any instance of LPN. The algorithm performs favorably compared to previously known algorithms, breaking the 80-bit security of the widely used (512,1/8) instance. The second one focuses on LPN instances over a polynomial ring, when the generator polynomial is reducible. Using the algorithm, we break an 80-bit security instance of the Lapin cryptosystem
Transceiver Design with Iterative Decoding of Capacity-Approaching codes over Fading channels
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