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Second-order asymptotics in fixed-length source coding and intrinsic randomness

IEEE TRANSACTIONS ON INFORMATION THEORY 2008年第10期54卷 4619-4637页

作者： Hayashi, Masahito JST ERATO Quantum Computat & Informat Project Tokyo 1130033 Japan

There is a difference between the optimal rates of fixed-length source coding and intrinsic randomness when we care about the second-order asymptotics. We prove this difference for general information sources and then investigate independent and identically distributed (i.i.d.) random variables and Markovian variables as examples. The difference is demonstrated through an investigation of the second-order asymptotic behavior of the rates. A universal fixed-length source code attaining the second-order optimal rate is also proposed. The difference between the rates of fixed-length source coding and intrinsic randomness proves that the outputs of fixed-length source codes are not uniformly distributed.

关键词： fixed-length source coding information spectrum intrinsic randomness second-order asymptotics source coding folklore

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coding theorems on the threshold scheme for a general source

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IEEE TRANSACTIONS ON INFORMATION THEORY 2008年第6期54卷 2658-2677页

作者： Koga, Hiroki Univ Tsukuba Grad Sch Syst & Informat Engn Tsukuba Ibaraki 3058573 Japan

In this paper, coding theorems on the (t, m)-threshold scheme for a general source are discussed, where m. means the number of the shares and t means a threshold. The (t, m)-threshold scheme treated in this paper encrypts n source outputs X-n to m. shares at once and is required to satisfy the two conditions that 1) X-n is reproduced from arbitrary t shares, and 2) almost no information of X-n is revealed from any t - 1 shares. It is shown that the (t, m)-threshold scheme must satisfy certain inequalities including the limit inferiors in probability. One of the inequalities is closely related to the minimum length of the fair random bits needed to a dealer for realizing the (t, m)-threshold scheme. In addition, it is shown that a certain variation of Shamir's threshold scheme meets the two conditions. The same approach can be taken to the problems of Shannon's cipher system with the perfect secrecy and fixed-length source coding with vanishing decoding error probability. It is shown that the same kind of inequalities, which indicate the converse coding theorems, hold in both two cases.

关键词： fixed-length source coding information-spectrum methods secret sharing scheme Shannon's cipher system threshold scheme

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coding theorems on the threshold scheme for a general source

Coding theorems on the threshold scheme for a general source

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IEEE Information Theory Workshop

作者： Koga, Hiroki Univ Tsukuba Grad Sch Syst & Informat Engn Tsukuba Ibaraki 3058573 Japan

ISBN: (纸本)0780394917

关键词： fixed-length source coding information-spectrum methods secret sharing scheme Shannon's cipher system threshold scheme

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An information-spectrum approach to classical and quantum hypothesis testing for simple hypotheses

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IEEE TRANSACTIONS ON INFORMATION THEORY 2007年第2期53卷 534-549页

作者： Nagaoka, Hiroshi Hayashi, Masahito Univ Electrocommun Grad Sch Informat Syst Tokyo 1828585 Japan RIKEN Lab Math Neurosci Brain Sci Inst Saitama 3510198 Japan

The information-spectrum analysis made by Han for classical hypothesis testing for simple hypotheses is extended to a unifying framework including both classical and quantum hypothesis testing. The results are also applied to fixed-length source coding when loosening the normalizing condition for probability distributions and for quantum states. We establish general formulas for several quantities relating to the asymptotic optimality of tests/codes in terms of classical and quantum information spectra.

关键词： classical hypothesis testing fixed-length source coding information spectrum optimal exponent quantum hypothesis testing

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Probability of error for the fixed-length lossy coding of general sources

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IEEE TRANSACTIONS ON INFORMATION THEORY 2005年第4期51卷 1498-1507页

作者： Iriyama, K 1-3 Mukaiyama Chou Ichinomiya Aichi 4910869 Japan

In this correspondence, we are interested in the error exponent of fixed-length lossy source coding, where the sources are general sources, in the sense of Han and Verdu, including all nonstationary and/or nonergodic sources. The aim of the correspondence is to establish a unified formula for the minimum (D, r) -achievable rate which is the minimum achievable coding rate under asymptotic constraints of the form epsilon(n)(D) similar to e(-nr) (n -> infinity), where r is the prescribed error exponent, epsilon(n) (D) is the probability of the distortion exceeding a level D, and n is the code-length. For the stationary memoryless source with finite alphabet, Marton (1974) obtained a formula for the reliability function which is expressed in terms of the minimum (D, r) -achievable rate. Recently, Ihara and Kubo proved that the Marton's formula remains true for the stationary memoryless Gaussian source under a mean-squared fidelity criterion. In this correspondence, it is shown that a formula similar to Marton's formula holds for the general sources. The error exponent of correct decoding is also investigated and a formula for the minimum achievable rate of crrect decoding in lossy coding is established.

关键词： continuous alphabet error exponent fixed-length source coding general source information spectrum large-deviation rate-distortion theory reliability function

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Probability of error for the fixed-length source coding of general sources

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IEEE TRANSACTIONS ON INFORMATION THEORY 2001年第4期47卷 1537-1543页

作者： Iriyama, K Nagoya Univ Grad Sch Human Informat Nagoya Aichi 4648601 Japan

We consider general sources in the sense of Han and Verdu. The information spectrum approach developed by Hen and Verdu has played a central role in the analyses of general sources. We investigate the asymptotic behavior of the probability of error for the fixed-length source coding of general sources. Recently, Han characterized the minimum r-achievable fixed-length coding rate. Our aim is to give an alternative expression for the minimum r-achievable rate.

关键词： countably infinite alphabet error exponent fixed-length source coding general source information spectrum large deviation

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Csiszar's cutoff rates for arbitrary discrete sources

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IEEE TRANSACTIONS ON INFORMATION THEORY 2001年第1期47卷 330-338页

作者： Chen, PN Alajaji, F Natl Chiao Tung Univ Dept Commun Engn Hsinchu Taiwan Queens Univ Dept Math & Stat Kingston ON K7L 3N6 Canada

Csiszar's forward beta -cutoff rate (given a fixed beta > 0) for a discrete source is defined as the smallest number Ro such that for every R > R-0, there exists a sequence of tired-length codes of rate R with probability of error asymptotically vanishing as e(-n beta (R-R0)). For a discrete memoryless source (DMS), the forward beta -cutoff rate is shown by Csiszar [6] to be equal to the source Renyi entropy. An analogous concept of reverse beta -cutoff rate regarding the probability of correct decoding is also characterized by Csiszar in terms of the Renyi entropy. In this work, Csiszar's results are generalized by investigating the beta -cutoff rates for the class of arbitrary discrete sources with memory. It is demonstrated that the limsup and liminf Renyi entropy rates provide the formulas for the forward and reverse beta -cutoff rates, respectively. Consequently, new fixed-length source coding operational characterizations for the Renyi entropy rates are established.

关键词： arbitrary sources with memory cutoff rates fixed-length source coding probability of error Renyi's entropy rates source reliability function

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The reliability functions of the general source with fixed-length coding

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IEEE TRANSACTIONS ON INFORMATION THEORY 2000年第6期46卷 2117-2132页

作者： Han, TS Univ Electrocommun Grad Sch Informat Syst Chofu Tokyo 1828585 Japan

The reliability function problems with fixed-length source coding for the general source are studied for all rates R. Our fundamental philosophy in doing so is to convert all of the reliability function problems to the pertinent computation problems in the large deviation-probability theory. It turns out that this kind of new methodology, which was recently developed by Han, enables us to establish quite compact general formulas of the reliability function for general sources including all nonstationary and/or nonergodic sources with countably infinite alphabet, Such general formulas are presented from the information-spectrum point of view.

关键词： countably infinite alphabet fixed-length source coding general source information spectrum information spectrum slicing large deviation reliability function

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Weak variable-length source coding

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IEEE TRANSACTIONS ON INFORMATION THEORY 2000年第4期46卷 1217-1226页

作者： Han, TS Univ Electrocommun Grad Sch Informat Syst Chofu Tokyo 1828585 Japan

Given a general source X = {X-n}(n=1)(infinity), source coding is characterized by a pair (phi(n), psi(n)) of encoder phi(n) and decoder psi(n) together with the probability of error epsilon(n) = Pr{psi(n)(phi(n)(X-n)) not equal X-n}. If the length of the encoder output phi(n)(X-n) is fixed, then it is called fixed-length source coding, while if the length of the encoder output phi(n)(X-n) is variable, then it is called variable-length source coding. Usually, in the context of fixed-length source coding the probability of error epsilon(n) is required to asymptotically vanish (i.e., lim(n-->infinity) epsilon(n) = 0), whereas in the context of variable-length source coding the probability of error epsilon(n) is required to be exactly zero (i.e., epsilon(n) = 0 For All n = 1, 2,....) In contrast to these, we consider in the present paper the problem of variable-length source coding with asymptotically vanishing probability of error (i.e., lim(n-->infinity) epsilon(n) = 0), and establish several fundamental theorems on this new subject.

关键词： countably infinite source alphabet fixed-length source coding general source uniform integrability variable-length source coding weak sup-entropy rate weak variable-length source coding

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The role of the asymptotic equipartition property in noiseless source coding

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IEEE TRANSACTIONS ON INFORMATION THEORY 1997年第3期43卷 847-857页

作者： Verdu, S Han, TS UNIV ELECTROCOMMUN GRAD SCH INFORMAT SYST CHOFU TOKYO 182 JAPAN

The (noiseless) fixed-length source coding theorem states that, except for outcomes in a set of vanishing probability, a source can be encoded at its entropy but not more efficiently, It is well known that the Asymptotic Equipartition Property (AEP) is a sufficient condition for a source to be encodable at its entropy, This paper shows that the AEP is necessary for the source coding theorem to hold for nonzero-entropy finite-alphabet sources, Furthermore, we show that a nonzero-entropy finite-alphabet source satisfies the direct coding theorem if and only if it satisfies the strong converse, In addition, we introduce the more general setting of nonserial information sources which need not put out strings of symbols, In this context, which encompasses the conventional serial setting, the AEP is equivalent to the validity of the strong coding theorem, Fundamental limits for data compression of nonserial information sources are shown based on the flat-top property-a new sufficient condition for the AEP.

关键词： asymptotic equipartition property entropy fixed-length source coding noiseless data compression Shannon theory source coding theorem

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