The Delsarte linear program is used to bound the size of codes given their block length n and minimal distance d by taking a linear relaxation from codes to quasicodes. We study for which values of (n, d) this linear ...
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The Delsarte linear program is used to bound the size of codes given their block length n and minimal distance d by taking a linear relaxation from codes to quasicodes. We study for which values of (n, d) this linear program has a unique optimum: while we show that it does not always have a unique optimum, we prove that it does if d > n/2 or if d <= 2. Introducing the Krawtchouk decomposition of a quasicode, we prove there exist optima to the (n, 2e) and (n - 1, 2e - 1) linear programs that have essentially identical Krawtchouk decompositions, revealing a parity phenomenon among the Delsarte linear programs. We generalize the notion of extending and puncturing codes to quasicodes, from which we see that this parity relationship is given by extending/puncturing. We further characterize these pairs of optima, in particular demonstrating that they exhibit a symmetry property, effectively halving the number of decision variables.
In this paper, we give several new constructions of write-once-memory (WOM) codes. The novelty in our constructions is the use of the so-called Wozencraft ensemble of linear codes. Specifically, we obtain the followin...
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In this paper, we give several new constructions of write-once-memory (WOM) codes. The novelty in our constructions is the use of the so-called Wozencraft ensemble of linear codes. Specifically, we obtain the following results. We give an explicit construction of a two-write WOM code that approaches capacity, over the binary alphabet. More formally, for every epsilon > 0, 0 < p < 1, and n = (1/epsilon)(O(1/p epsilon)), we give a construction of a two-write WOM code of length n and capacity H(p) + 1 - p - epsilon. Since the capacity of a two-write WOM code is max(p)(H(p) + 1 - p), we get a code that is epsilon-close to capacity. Furthermore, encoding and decoding can be done in time O(n(2) . poly(log n)) and time O(n . poly(log n)), respectively, and in logarithmic space. In addition, we exhibit an explicit randomized encoding scheme of a two-write capacity-achieving WOM code of block length polynomial in 1/epsilon (again, epsilon is the gap to capacity), with a polynomial time encoding and decoding. We obtain a new encoding scheme for three-write WOM codes over the binary alphabet. Our scheme achieves rate 1.809 - epsilon, when the block length is exp(1/epsilon). This gives a better rate than what could be achieved using previous techniques. We highlight a connection to linear seeded extractors for bit-fixing sources. In particular, we show that obtaining such an extractor with seed length O(log n) can lead to improved parameters for two-write WOM codes. We then give an application of existing constructions of extractors to the problem of designing encoding schemes for memory with defects.
The error probability of maximum-likelihood (ML) soft-decision decoded binary block codes rarely accepts exact closed forms. In addition, for long codes ML decoding becomes prohibitively complex. Nevertheless, bounds ...
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The error probability of maximum-likelihood (ML) soft-decision decoded binary block codes rarely accepts exact closed forms. In addition, for long codes ML decoding becomes prohibitively complex. Nevertheless, bounds on the performance of ML decoded systems provide insight into the effect of system parameters on the overall system performance in addition to a measure of efficiency of the sub-optimum decoding methods used in practice. In the article, a comprehensive study of a number of lower and tipper bounds on the error probability of ML decoding of binary codes over AWGN channel is provided. Bounds considered here are bounds based oil the so-called Bonferroni-type inequalities and bounds developed primarily in the light of the geometrical structure of the underlying signal constellations. The interrelationships among the bounds are explored and current tightest bounds at different noise levels are pointed out.
In this paper we translate in terms of coding theory constraints that are used in designing DNA codes for use in DNA computing or as bar-codes in chemical libraries. We propose new constructions for DNA codes satisfyi...
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In this paper we translate in terms of coding theory constraints that are used in designing DNA codes for use in DNA computing or as bar-codes in chemical libraries. We propose new constructions for DNA codes satisfying either a reverse-complement constraint, a GC-content constraint, or both, that are derived from additive and linear codes over four-letter alphabets. We focus in particular on codes over GF(4), and we construct new DNA codes that are in many cases better (sometimes far better) than previously known codes. We provide updated tables up to length 20 that include these codes as well as new codes constructed using a combination of lexicographic techniques and stochastic search. (c) 2004 Elsevier B.V. All rights reserved.
The study of self-testing and self-correcting programs leads to the search for robust characterizations of functions. Here the authors make this notion precise and show such a characterization for polynomials. From th...
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The study of self-testing and self-correcting programs leads to the search for robust characterizations of functions. Here the authors make this notion precise and show such a characterization for polynomials. From this characterization, the authors get the following applications. Simple and efficient self-testers for polynomial functions are constructed. The characterizations provide results in the area of coding theory by giving extremely fast and efficient error-detecting schemes for some well-known codes. This error-detection scheme plays a crucial role in subsequent results on the hardness of approximating some NP-optimization problems.
We present an investigation of epsilon -entropy, h(epsilon), in dynamical systems, stochastic processes and turbulence, This tool allows for a suitable characterization of dynamical behaviours arising in systems with ...
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We present an investigation of epsilon -entropy, h(epsilon), in dynamical systems, stochastic processes and turbulence, This tool allows for a suitable characterization of dynamical behaviours arising in systems with many different scales of motion. Particular emphasis is put on a recently proposed approach to the calculation of the epsilon -entropy based on the exit-time statistics. The advantages of this method are demonstrated in examples of deterministic diffusive maps, intermittent maps, stochastic self- and multi-affine signals and experimental turbulent data. Concerning turbulence, the multifractal formalism applied to the exit-time statistics allows us to predict that h(epsilon) similar to epsilon (-3) for velocity-time measurement. This power law is independent of the presence of intermittency and has been confirmed by the experimental data analysis. Moreover, we show that the epsilon -entropy density of a three-dimensional velocity field is affected by the correlations induced by the sweeping of large scales. (C) 2000 Elsevier Science B.V. All rights reserved.
Ant colony system (ACS) has been widely applied for solving discrete domain problems in recent years. In particular, they are efficient and effective in finding nearly optimal solutions to discrete search spaces. Beca...
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Ant colony system (ACS) has been widely applied for solving discrete domain problems in recent years. In particular, they are efficient and effective in finding nearly optimal solutions to discrete search spaces. Because of the restriction of ant-based algorithms, when the solution space of a problem to be solved is continuous, it is not so appropriate to use the original ACS to solve it. However, engineering mathematics in the real applications are always applied in the continuous domain. This paper thus proposes an extended ACS approach based on binary-coding to provide a standard process for solving problems with continuous variables. It first encodes solution space for continuous domain into a discrete binary-coding space (searching map), and a modified ACS can be applied to find the solution. Each selected edge in a complete path represents a part of a candidate solution. Different from the previous ant-based algorithms for continuous domain, the proposed binary coding ACS (BCACS) could retain the original operators and keep the benefits and characteristics of the traditional ACS. Besides, the proposed approach is easy to implement and could be applied in different kinds of problems in addition to mathematical problems. Several constrained functions are also evaluated to demonstrate the performance of the proposed algorithm.
The article discusses the management and coding of distributed computer systems and asynchronous programming. The author reflects on the use of an abstraction known as promises to simplify asynchronous function call m...
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The article discusses the management and coding of distributed computer systems and asynchronous programming. The author reflects on the use of an abstraction known as promises to simplify asynchronous function call management and parallel processing. Other topics include the writing of concurrent processing systems, the algorithmic design of promises, the central authority of the *** process, health checks for processing workers, and synchronization among central authorities in multiple servers or processes.
It is known that the set of all primitive words and the set of all -primitive words are disjunctive languages. In this paper we prove that the set of all -primitive words is disjunctive. We also prove that the set of ...
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It is known that the set of all primitive words and the set of all -primitive words are disjunctive languages. In this paper we prove that the set of all -primitive words is disjunctive. We also prove that the set of all primitive but not -primitive words, the set of all balanced but not -primitive words, and the set of all -primitive but not -primitive words are disjunctive languages.
Coded computing is a new framework to address fundamental issues in large scale distributed computing, by injecting structured randomness and redundancy. We first provide an overview of coded computing and summarize s...
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Coded computing is a new framework to address fundamental issues in large scale distributed computing, by injecting structured randomness and redundancy. We first provide an overview of coded computing and summarize some recent advances. Then we focus on distributed matrix multiplication and consider a common scenario where each worker is assigned a fraction of the multiplication task. In particular, by partitioning two input matrices into m-by-p and p-by-n subblocks, a single multiplication task can be viewed as computing linear combinations of pmn submatrix products, which can be assigned to pmn workers. Such block-partitioning-based designs have been widely studied under the topics of secure, private, and batch computation, where the state of the arts all require computing at least "cubic" (pmn) number of submatrix multiplications. Entangled polynomial codes, first presented for straggler mitigation, provides a powerful method for breaking the cubic barrier. It achieves a subcubic recovery threshold, i.e., recovering the final product from any subset of multiplication results with a size order-wise smaller than pmn. We show that entangled polynomial codes can be further extended to also include these three important settings, providing unified frameworks that order-wise reduce the total computational costs by achieving subcubic recovery thresholds.
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