We put forward an ample framework for coding based on upper probabilities, or more generally on normalized monotone set-measures, and model accordingly noisy transmission channels and decoding errors. Two inverse prob...
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We put forward an ample framework for coding based on upper probabilities, or more generally on normalized monotone set-measures, and model accordingly noisy transmission channels and decoding errors. Two inverse problems are considered. In the first case, a decoder is given and one looks for channels of a specified family over which that decoder would work properly. In the second and more ambitious case, it is codes which are given, and one looks for channels over which those codes would ensure the required error correction capabilities. Upper probabilities allow for a solution of the two inverse problems in the case of usual codes based on checking Hamming distances between codewords: one can equivalently check suitable upper probabilities of the decoding errors. This soon extends to "odd" codeword distances for DNA strings as used in DNA word design, where instead, as we prove, not even the first unassuming inverse problem admits of a solution if one insists on channel models based on "usual" probabilities.
In this paper, we introduce a new Fibonacci G(p,m) matrix for the m-extension of the Fibonacci p-numbers where p (>= 0) is integer and m (> 0). Thereby, we discuss various properties of G(p,m) matrix and the cod...
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In this paper, we introduce a new Fibonacci G(p,m) matrix for the m-extension of the Fibonacci p-numbers where p (>= 0) is integer and m (> 0). Thereby, we discuss various properties of G(p,m) matrix and the coding theory followed from the G(p,m) matrix. In this paper, we establish the relations among the code elements for all values of p (nonnegative integer) and m (> 0). We also show that the relation, among the code matrix elements for all values of p and m = 1, coincides with the relation among the code matrix elements for all values of p [Basu M, Prasad B. The generalized relations among the code elements for Fibonacci coding theory. Chaos, Solitons and Fractals (2008). doi: 10.1016/***.2008.09.030]. In general, correct ability of the method increases as p increases but it is independent of m. (c) 2009 Published by Elsevier Ltd.
Our research explores the feasibility of using communication theory, error control (EC) coding theory specifically, for quantitatively modeling the protein translation initiation mechanism. The messenger RNA (mRNA) of...
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Our research explores the feasibility of using communication theory, error control (EC) coding theory specifically, for quantitatively modeling the protein translation initiation mechanism. The messenger RNA (mRNA) of Escherichia coli K-12 is modeled as a noisy (errored), encoded signal and the ribosome as a minimum Hamming distance decoder, where the 16S ribosomal RNA (rRNA) serves as a template for generating a set of valid codewords (the codebook). We tested the E. coli based coding models on 5' untranslated leader sequences of prokaryotic organisms of varying taxonomical relation to E. coli including: Salmonella typhimurium LT2, Bacillus subtilis, and Staphylococcus aureus Mu50. The model identified regions on the 5' untranslated leader where the minimum Hamming distance values of translated mRNA sub-sequences and non-translated genomic sequences differ the most. These regions correspond to the Shine-Dalgarno domain and the non-random domain. Applying the EC coding-based models to B. subtilis, and S. aureus Mu50 yielded results similar to those for E. coli K-12. Contrary to our expectations, the behavior of S. typhimurium LT2, the more taxonomically related to E. coli, resembled that of the non-translated sequence group. (C) 2004 Elsevier Ireland Ltd. All rights reserved.
For fixed integers r >= 3,e >= 3,v >= r 1, an r-uniform hypergraph is called G tau(v, e)-free if the union of any e distinct edges contains at least v + 1 vertices. Brown, Erdos, and Sos showed that the maxim...
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For fixed integers r >= 3,e >= 3,v >= r 1, an r-uniform hypergraph is called G tau(v, e)-free if the union of any e distinct edges contains at least v + 1 vertices. Brown, Erdos, and Sos showed that the maximum number of edges of such a hypergraph on n vertices, denoted as f(r)(n, v, e), satisfies Omega(n (er-v/e-1)) = f(r)(n, v, e) = O(n(vertical bar er-v/e-1 vertical bar)). For sufficiently large n and e - 1 vertical bar er - v, the lower bound matches the upper bound up to a constant factor, which depends only on r, v, e;whereas for e - 1 inverted iota er - v, in general it is a notoriously hard problem to determine the correct exponent of n. Among other results, we improve the above lower bound by showing that f(r)(n, v, e) = Omega(n (er-v/e-1) (log n) (1/e-1)) for any r, e, v satisfying gcd(e - 1, er - v) = 1. The hypergraph we constructed is in fact G(r)(ir - inverted right perpendicular(i-1)(er-v)/e-1inverted left perpendicular, i)-free for every 2 <= i <= e, and it has several interesting applications in coding theory. The proof of the new lower bound is based on a novel application of the lower bound on the hypergraph independence number due to Duke, Lefmann, and Rodl.
In this paper, we present a new perspective of single server private information retrieval (PIR) schemes by using the notion of linear error-correcting codes. Many of the known single server schemes are based on takin...
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In this paper, we present a new perspective of single server private information retrieval (PIR) schemes by using the notion of linear error-correcting codes. Many of the known single server schemes are based on taking linear combinations between database elements and the query elements. Using the theory of linear codes, we develop a generic framework that formalizes all such PIR schemes. This generic framework provides an appropriate setup to analyze the security of such PIR schemes. In fact, we describe some known PIR schemes with respect to this code-based framework, and present the weaknesses of the broken PIR schemes in a unified point of view.
Ashared computer tool QPlus for studying coding theory studying is presented. The system offers computations over Z(q) = {0, 1, ... , q - 1} (q < 256) and includes modular arithmetic, elementary number theory, vect...
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Ashared computer tool QPlus for studying coding theory studying is presented. The system offers computations over Z(q) = {0, 1, ... , q - 1} (q < 256) and includes modular arithmetic, elementary number theory, vectors and matrices arithmetic and an environment for research on q-ary codes - linear, constant-weight and equidistant codes. QPlus includes a DLL library package that implements coding theory algorithms. We explore the problem of finding bounds on the size of q-ary codes by computer methods. Some examples for optimal equidistant codes and constant-weight equidistant codes that have been constructed by computer methods developed in QPlus are described. We also research some optimal linear codes.
We derive uniform asymptotic expressions of some Abel sums appearing in some problems in coding theory and indicate the usefulness of these sums in other fields, like empirical processes, machine maintenance, analysis...
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We derive uniform asymptotic expressions of some Abel sums appearing in some problems in coding theory and indicate the usefulness of these sums in other fields, like empirical processes, machine maintenance, analysis of algorithms, probabilistic number theory, queuing models, etc. (C) 2001 Elsevier Science B.V. All rights reserved.
We propose a method based on cluster expansion to study the optimal code with a given distance between codewords. Using this approach we find the Gilbert-Varshamov lower bound for the rate of largest code.
We propose a method based on cluster expansion to study the optimal code with a given distance between codewords. Using this approach we find the Gilbert-Varshamov lower bound for the rate of largest code.
We have considered a class of square Fibonacci matrix of order (p + 1) whose elements are based on the Fibonacci p numbers with determinant equal to +1 or -1. There is a relation between Fibonacci numbers with initial...
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We have considered a class of square Fibonacci matrix of order (p + 1) whose elements are based on the Fibonacci p numbers with determinant equal to +1 or -1. There is a relation between Fibonacci numbers with initial terms which is known as cassini formula. Fibonacci series and the golden mean plays a very important role in the construction of a relatively new space-time theory, which is known as E-infinity theory. An original Fibonacci coding/decoding method follows from the Fibonacci matrices. There already exists a relation between the code matrix elements for the case p = 1 [Stakhov AP. Fibonacci matrices, a generalization of the cassini formula and a new coding theory. Chaos, Solitons and Fractals 2006;30:56-66.]. In this paper, we have established generalized relations among the code matrix elements for all values of p. For p = 2, the correct ability of the method is 99.80%. In general, correct ability of the method increases as p increases. (C) 2008 Elsevier Ltd. All rights reserved.
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