2,695 research outputs found
Artin's primitive root conjecture -a survey -
This is an expanded version of a write-up of a talk given in the fall of 2000
in Oberwolfach. A large part of it is intended to be understandable by
non-number theorists with a mathematical background. The talk covered some of
the history, results and ideas connected with Artin's celebrated primitive root
conjecture dating from 1927. In the update several new results established
after 2000 are also discussed.Comment: 87 pages, 512 references, to appear in Integer
Optimal Partitioned Cyclic Difference Packings for Frequency Hopping and Code Synchronization
Optimal partitioned cyclic difference packings (PCDPs) are shown to give rise
to optimal frequency-hopping sequences and optimal comma-free codes. New
constructions for PCDPs, based on almost difference sets and cyclic difference
matrices, are given. These produce new infinite families of optimal PCDPs (and
hence optimal frequency-hopping sequences and optimal comma-free codes). The
existence problem for optimal PCDPs in , with base blocks
of size three, is also solved for all .Comment: to appear in IEEE Transactions on Information Theor
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
Pruned Bit-Reversal Permutations: Mathematical Characterization, Fast Algorithms and Architectures
A mathematical characterization of serially-pruned permutations (SPPs)
employed in variable-length permuters and their associated fast pruning
algorithms and architectures are proposed. Permuters are used in many signal
processing systems for shuffling data and in communication systems as an
adjunct to coding for error correction. Typically only a small set of discrete
permuter lengths are supported. Serial pruning is a simple technique to alter
the length of a permutation to support a wider range of lengths, but results in
a serial processing bottleneck. In this paper, parallelizing SPPs is formulated
in terms of recursively computing sums involving integer floor and related
functions using integer operations, in a fashion analogous to evaluating
Dedekind sums. A mathematical treatment for bit-reversal permutations (BRPs) is
presented, and closed-form expressions for BRP statistics are derived. It is
shown that BRP sequences have weak correlation properties. A new statistic
called permutation inliers that characterizes the pruning gap of pruned
interleavers is proposed. Using this statistic, a recursive algorithm that
computes the minimum inliers count of a pruned BR interleaver (PBRI) in
logarithmic time complexity is presented. This algorithm enables parallelizing
a serial PBRI algorithm by any desired parallelism factor by computing the
pruning gap in lookahead rather than a serial fashion, resulting in significant
reduction in interleaving latency and memory overhead. Extensions to 2-D block
and stream interleavers, as well as applications to pruned fast Fourier
transforms and LTE turbo interleavers, are also presented. Moreover,
hardware-efficient architectures for the proposed algorithms are developed.
Simulation results demonstrate 3 to 4 orders of magnitude improvement in
interleaving time compared to existing approaches.Comment: 31 page
In-silico prediction of disorder content using hybrid sequence representation
<p>Abstract</p> <p>Background</p> <p>Intrinsically disordered proteins play important roles in various cellular activities and their prevalence was implicated in a number of human diseases. The knowledge of the content of the intrinsic disorder in proteins is useful for a variety of studies including estimation of the abundance of disorder in protein families, classes, and complete proteomes, and for the analysis of disorder-related protein functions. The above investigations currently utilize the disorder content derived from the per-residue disorder predictions. We show that these predictions may over-or under-predict the overall amount of disorder, which motivates development of novel tools for direct and accurate sequence-based prediction of the disorder content.</p> <p>Results</p> <p>We hypothesize that sequence-level aggregation of input information may provide more accurate content prediction when compared with the content extracted from the local window-based residue-level disorder predictors. We propose a novel predictor, DisCon, that takes advantage of a small set of 29 custom-designed descriptors that aggregate and hybridize information concerning sequence, evolutionary profiles, and predicted secondary structure, solvent accessibility, flexibility, and annotation of globular domains. Using these descriptors and a ridge regression model, DisCon predicts the content with low, 0.05, mean squared error and high, 0.68, Pearson correlation. This is a statistically significant improvement over the content computed from outputs of ten modern disorder predictors on a test dataset with proteins that share low sequence identity with the training sequences. The proposed predictive model is analyzed to discuss factors related to the prediction of the disorder content.</p> <p>Conclusions</p> <p>DisCon is a high-quality alternative for high-throughput annotation of the disorder content. We also empirically demonstrate that the DisCon's predictions can be used to improve binary annotations of the disordered residues from the real-value disorder propensities generated by current residue-level disorder predictors. The web server that implements the DisCon is available at <url>http://biomine.ece.ualberta.ca/DisCon/</url>.</p
Predicting the protein-protein interactions using primary structures with predicted protein surface
<p>Abstract</p> <p>Background</p> <p>Many biological functions involve various protein-protein interactions (PPIs). Elucidating such interactions is crucial for understanding general principles of cellular systems. Previous studies have shown the potential of predicting PPIs based on only sequence information. Compared to approaches that require other auxiliary information, these sequence-based approaches can be applied to a broader range of applications.</p> <p>Results</p> <p>This study presents a novel sequence-based method based on the assumption that protein-protein interactions are more related to amino acids at the surface than those at the core. The present method considers surface information and maintains the advantage of relying on only sequence data by including an accessible surface area (ASA) predictor recently proposed by the authors. This study also reports the experiments conducted to evaluate a) the performance of PPI prediction achieved by including the predicted surface and b) the quality of the predicted surface in comparison with the surface obtained from structures. The experimental results show that surface information helps to predict interacting protein pairs. Furthermore, the prediction performance achieved by using the surface estimated with the ASA predictor is close to that using the surface obtained from protein structures.</p> <p>Conclusion</p> <p>This work presents a sequence-based method that takes into account surface information for predicting PPIs. The proposed procedure of surface identification improves the prediction performance with an <it>F-measure </it>of 5.1%. The extracted surfaces are also valuable in other biomedical applications that require similar information.</p
Error-Correction Coding and Decoding: Bounds, Codes, Decoders, Analysis and Applications
Coding; Communications; Engineering; Networks; Information Theory; Algorithm
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