10,509 research outputs found
Using Cantor Sets for Error Detection
Error detection is a fundamental need in most computer networks and communication systems in order to combat the effect of noise. Error detection techniques have also been incorporated with lossless data compression algorithms for transmission across communication networks. In this paper, we propose to incorporate a novel error detection scheme into a Shannon optimal lossless data compression algorithm known as Generalized Luröth Series (GLS) coding. GLS-coding is a generalization of the popular Arithmetic Coding which is an integral part of the JPEG2000 standard for still image compression. GLS-coding encodes the input message as a symbolic sequence on an appropriate 1D chaotic map Generalized Luröth Series (GLS) and the compressed file is obtained as the initial value by iterating backwards on the
map. However, in the presence of noise, even small errors in the compressed file leads to catastrophic decoding errors owing to sensitive dependence on initial values, the hallmark of deterministic chaos. In this paper, we first show that repetition codes, the oldest and the most basic error correction and detection codes in literature, actually lie on a Cantor set with a fractal dimension of 1n1n, which is also the rate of the code. Inspired by this, we incorporate error detection capability to GLS-coding by ensuring that the compressed file (initial value on the chaotic map) lies on a Cantor set. Even a 1-bit error in the initial value will throw it outside the Cantor set, which can be detected while decoding. The rate of the code can be adjusted by the fractal dimension of the Cantor set, thereby controlling the error detection performance
Fast Decoder for Overloaded Uniquely Decodable Synchronous Optical CDMA
In this paper, we propose a fast decoder algorithm for uniquely decodable
(errorless) code sets for overloaded synchronous optical code-division
multiple-access (O-CDMA) systems. The proposed decoder is designed in a such a
way that the users can uniquely recover the information bits with a very simple
decoder, which uses only a few comparisons. Compared to maximum-likelihood (ML)
decoder, which has a high computational complexity for even moderate code
lengths, the proposed decoder has much lower computational complexity.
Simulation results in terms of bit error rate (BER) demonstrate that the
performance of the proposed decoder for a given BER requires only 1-2 dB higher
signal-to-noise ratio (SNR) than the ML decoder.Comment: arXiv admin note: substantial text overlap with arXiv:1806.0395
Fractal states in quantum information processing
The fractal character of some quantum properties has been shown for systems
described by continuous variables. Here, a definition of quantum fractal states
is given that suits the discrete systems used in quantum information
processing, including quantum coding and quantum computing. Several important
examples are provided
Extreme Value distribution for singular measures
In this paper we perform an analytical and numerical study of Extreme Value
distributions in discrete dynamical systems that have a singular measure. Using
the block maxima approach described in Faranda et al. [2011] we show that,
numerically, the Extreme Value distribution for these maps can be associated to
the Generalised Extreme Value family where the parameters scale with the
information dimension. The numerical analysis are performed on a few low
dimensional maps. For the middle third Cantor set and the Sierpinskij triangle
obtained using Iterated Function Systems, experimental parameters show a very
good agreement with the theoretical values. For strange attractors like Lozi
and H\`enon maps a slower convergence to the Generalised Extreme Value
distribution is observed. Even in presence of large statistics the observed
convergence is slower if compared with the maps which have an absolute
continuous invariant measure. Nevertheless and within the uncertainty computed
range, the results are in good agreement with the theoretical estimates
Fractal Strings and Multifractal Zeta Functions
For a Borel measure on the unit interval and a sequence of scales that tend
to zero, we define a one-parameter family of zeta functions called multifractal
zeta functions. These functions are a first attempt to associate a zeta
function to certain multifractal measures. However, we primarily show that they
associate a new zeta function, the topological zeta function, to a fractal
string in order to take into account the topology of its fractal boundary. This
expands upon the geometric information garnered by the traditional geometric
zeta function of a fractal string in the theory of complex dimensions. In
particular, one can distinguish between a fractal string whose boundary is the
classical Cantor set, and one whose boundary has a single limit point but has
the same sequence of lengths as the complement of the Cantor set. Later work
will address related, but somewhat different, approaches to multifractals
themselves, via zeta functions, partly motivated by the present paper.Comment: 32 pages, 9 figures. This revised version contains new sections and
figures illustrating the main results of this paper and recent results from
others. Sections 0, 2, and 6 have been significantly rewritte
Languages of lossless seeds
Several algorithms for similarity search employ seeding techniques to quickly
discard very dissimilar regions. In this paper, we study theoretical properties
of lossless seeds, i.e., spaced seeds having full sensitivity. We prove that
lossless seeds coincide with languages of certain sofic subshifts, hence they
can be recognized by finite automata. Moreover, we show that these subshifts
are fully given by the number of allowed errors k and the seed margin l. We
also show that for a fixed k, optimal seeds must asymptotically satisfy l ~
m^(k/(k+1)).Comment: In Proceedings AFL 2014, arXiv:1405.527
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