A sufficient condition for a code to be optimum on discrete channels with finite input and output alphabets is given, where being optimum means achieving the minimum decoding error probability. This condition is deriv...
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A sufficient condition for a code to be optimum on discrete channels with finite input and output alphabets is given, where being optimum means achieving the minimum decoding error probability. This condition is derived by generalizing the ideas of binary perfect and quasi-perfect codes, which are known to be optimum on the binary symmetric channel. An application of the sufficient condition shows that the code presented by Hamada and Fujiwara (1997) is optimum on the q-ary channel model proposed by Fuja and Heegard (1990), where q is a prime power with some restriction. The channel model is subject to two types of additive errors of (in general) different probabilities.
The necessary and sufficient condition for constructing a spreading set with decodability is investigated. It is proved that for a given delta-decodable spreading set and a q X q square matrix H with components 1 or -...
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The necessary and sufficient condition for constructing a spreading set with decodability is investigated. It is proved that for a given delta-decodable spreading set and a q X q square matrix H with components 1 or -1, a q delta-decodable spreading set S* is obtained if and only if H is a Hadamard matrix. In addition, a decoding rule with error correction and message data detection is provided.
The main construction for resilient functions uses linear error-correcting codes;a resilient function constructed in this way is said to be linear. It has been conjectured that if a resilient function exists, then a l...
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The main construction for resilient functions uses linear error-correcting codes;a resilient function constructed in this way is said to be linear. It has been conjectured that if a resilient function exists, then a linear function with the same parameters exists. In this note we construct infinite classes of nonlinear resilient functions from the Kerdock and Preparata codes. We also show that linear resilient functions having the same parameters as the functions that we construct from the Kerdock codes do not exist. Thus, the aforementioned conjecture is disproved.
New mathematical techniques for analysis of raw dumps of NAND flash memory were developed. These techniques are aimed at detecting, by analysis of the raw NAND flash dump only, the use of LFSR-based scrambling and the...
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New mathematical techniques for analysis of raw dumps of NAND flash memory were developed. These techniques are aimed at detecting, by analysis of the raw NAND flash dump only, the use of LFSR-based scrambling and the use of a binary cyclic code for error-correction. If detected, parameter values for both LFSR and cyclic error-correcting code are determined simultaneously. These can subsequently be applied to expose the content of memory pages in the raw NAND flash dump and prepare these for further processing with media analysis tools. The techniques were tested on raw NAND flash memory dumps of four different devices and in all cases LFSR-based scrambling and binary cyclic error-correcting codes were in use. (C) 2015 Elsevier Ltd. All rights reserved.
A coding scheme of (k + 1)-ary error-correcting signature codes for a noisy multiple-access adder channel is proposed. Given a signature matrix A and a difference matrix D = D+ - D- a priori, a larger signature matrix...
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A coding scheme of (k + 1)-ary error-correcting signature codes for a noisy multiple-access adder channel is proposed. Given a signature matrix A and a difference matrix D = D+ - D- a priori, a larger signature matrix is obtained by replacing each element in the Hadamard matrix with A, or D+, or D- depending on the values of the elements and their locations in the Hadamard matrix. The set of rows in the proposed matrix gives an error-correcting signature code. Introducing a difference matrix makes it possible to construct an error-correcting signature code whose sum rate is increased with an increase in the order of the Hadamard matrix. Either binary or non-binary signature codes are constructed when the pairs of matrices A and D are given.
We show that unbounded number of channel uses may be necessary for perfect transmission of quantum information. For any n, we explicitly construct low-dimensional quantum channels (input dimension 4, Choi rank 2 or 4)...
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We show that unbounded number of channel uses may be necessary for perfect transmission of quantum information. For any n, we explicitly construct low-dimensional quantum channels (input dimension 4, Choi rank 2 or 4) whose quantum zero-error capacity is positive, but the corresponding n-shot capacity is zero. We give estimates for quantum zero-error capacity of such channels as a function of n and show that these channels can be chosen in any small vicinity (in the -norm) of a classical-quantum channel. Mathematically, this property means appearance of an ideal (noiseless) subchannel only in sufficiently large tensor power of a channel. Our approach (using special continuous deformation of a maximal commutative -subalgebra of ) also gives low-dimensional examples of the superactivation of 1-shot quantum zero-error capacity. Finally, we consider multi-dimensional construction which increases the estimate for quantum zero-error capacity of channels having vanishing n-shot capacity.
With scaling of process technologies and increase in process variations, embedded memories will be inherently unreliable. Approximate computing is a new class of techniques that relax the accuracy requirement of compu...
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With scaling of process technologies and increase in process variations, embedded memories will be inherently unreliable. Approximate computing is a new class of techniques that relax the accuracy requirement of computing systems. In this paper, we present the Adaptive Coding for approximate Computing (ACOCO) framework, which provides us with an analysis-guided design methodology to develop adaptive codes for different computations on the data read from faulty memories. In ACOCO, we first compress the data by introducing distortion in the source encoder, and then add redundant bits to protect the data against memory errors in the channel encoder. We are thus able to protect the data against memory errors without additional memory overhead so that the coded data have the same bit-length as the uncoded data. We design the source encoder by first specifying a cost function measuring the effect of the data compression on the system output, and then design the source code according to this cost function. We develop adaptive codes for two types of systems under ACOCO. The first type of systems we consider, which includes many machine learning and graph-based inference systems, is the systems dominated by product operations. We evaluate the cost function statistics for the proposed adaptive codes, and demonstrate its effectiveness via two application examples: max-product image denoising and naive Bayesian classification. Next, we consider another type of systems: iterative decoders with min operation and sign-bit decision, which are widely applied in wireless communication systems. We develop an adaptive coding scheme for the min-sum decoder subject to memory errors. A density evolution analysis and simulations on finite length codes both demonstrate that the decoder with our adaptive code achieves a residual error rate that is on the order of the square of the residual error rate achieved by the nominal min-sum decoder.
Let G be a finite graph with mu, as an eigenvalue of multiplicity k. A star set for mu. is a set X of k vertices in G such that mu is not an eigenvalue of G - X. We investigate independent star sets of largest possibl...
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Let G be a finite graph with mu, as an eigenvalue of multiplicity k. A star set for mu. is a set X of k vertices in G such that mu is not an eigenvalue of G - X. We investigate independent star sets of largest possible size in a variety of situations. We note connections with symmetric designs, codes, strongly regular graphs, and graphs with least eigenvalue -2. (C) 2013 Elsevier Inc. All rights reserved.
That semiosis is specific to the living world is the cornerstone of biosemiotics. For checking an information-theoretic interpretation of this statement already proposed at the 2009 Biosemiotics Gathering in Prague, i...
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That semiosis is specific to the living world is the cornerstone of biosemiotics. For checking an information-theoretic interpretation of this statement already proposed at the 2009 Biosemiotics Gathering in Prague, it is first attempted here to answer a question asked by Kupiec and Sonigo in Ni Dieu ni Gene (2000) on what differentiates living objects and those resulting from a geophysical process. Similar questions were asked by Schrodinger in his essay What Is Life? where the emphasis was laid on the relationship of the atomic scale of the genes and the macroscopic scale of living beings. This essay was published in 1944, before information was introduced as a scientific entity, at a time when DNA was not yet identified as the vector of heredity. We undertake answering some of these questions, arguing that the living world is made of organisms, i.e., of assemblies possessing in a genome the information needed for their replication and their maintenance while the inanimate world only contains aggregates. In short, a biological process keeps its order through the use of information. For defining order, it is proposed that an orderly object can be produced by a construction (e.g., the copy of a template) using available data within some given context. In other words, replicating an orderly object does not bring new information into its context. Order in this meaning appears as specific to the living world, at variance with the inanimate world which is basically disorderly. A better understanding of what separates the living world from the inanimate world results: the use of information is the distinguishing feature which defines their border. Any living thing contains a symbolic information, referred to as its genome, inscribed into DNA molecules. This genome can indeed be copied but, its support being embedded in the physical world, it incurs disturbances which result in symbol errors. Keeping its order thus needs endowing any genome with error correction ability:
Let X be a finite set of q elements, and n, K, d be integers. A subset C subset of X(n) is an (n, K, d) error-correcting code, if #(C) = K and its minimum distance is d. We define an (n, K, d) error-correcting sequenc...
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Let X be a finite set of q elements, and n, K, d be integers. A subset C subset of X(n) is an (n, K, d) error-correcting code, if #(C) = K and its minimum distance is d. We define an (n, K, d) error-correcting sequence over X as a periodic sequence {a(i)}(i=0,1,...) (a(i) is an element of X) with period K, such that the set of all consecutive n-tuples of this sequence form an (n, K, d) error-correcting code over X. Under a moderate conjecture on the existence of some type of primitive polynomials, we prove that there is a (q(m)-1/q-1, q(qm-1/q-1-m) - 1,3) errorcorrecting sequence, such that its code-set is the q-ary Hamming code [q(m)-1/q-1, q(m)-1/q-1 -m, 3] with 0 removed, for q > 2 being a prime power. For the case q = 2, under a similar conjecture, we prove that there is a (2(m) -2, 2(2m-m-2) -1,3) error-correcting sequence, such that its code-set supplemented with 0 is the subset of the binary Hamming code [2(m) -1, 2(m) -1 -m, 3] obtained by requiring one specified coordinate being 0.
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