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19
New Upper Bounds on Error Exponents
"... We derive new upper bounds on the error exponents for the maximum likelihood decoding and error detecting in the binary symmetric channels. This is an improvement on the straightline bound by ShannonGallagerBerlekamp (1967) and the McElieceOmura (1977) minimum distance bound. For the probability ..."
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Cited by 28 (6 self)
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We derive new upper bounds on the error exponents for the maximum likelihood decoding and error detecting in the binary symmetric channels. This is an improvement on the straightline bound by ShannonGallagerBerlekamp (1967) and the McElieceOmura (1977) minimum distance bound. For the probability of undetected error the new bounds are better than the recent bound by AbdelGhaffar (1997) and the minimum distance and straightline bounds by Levenshtein (1978, 1989). We further extend the range of rates where the undetected error exponent is known to be exact. Keywords: Error exponents, Undetected error, Maximum likelihood decoding, Distance distribution, Krawtchouk polynomials. Submitted to IEEE Transactions on Information Theory 1 Introduction A classical problem of the information theory is to estimate probabilities of undetected and decoding errors when a block code is used for information transmission over a binary symmetric channel (BSC). We will study here exponential bounds ...
Universal Composite Hypothesis Testing: A Competitve Minimax Approach
, 2001
"... A novel approach is presented for the longstanding problem of composite hypothesis testing. In composite hypothesis testing, unlike in simple hypothesis testing, the probability function of the observed data given the hypothesis, is uncertain as it depends on the unknown value of some parameter. Th ..."
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Cited by 23 (7 self)
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A novel approach is presented for the longstanding problem of composite hypothesis testing. In composite hypothesis testing, unlike in simple hypothesis testing, the probability function of the observed data given the hypothesis, is uncertain as it depends on the unknown value of some parameter. The proposed approach is to minimize the worstcase ratio between the probability of error of a decision rule that is independent of the unknown parameters and the minimum probability of error attainable given the parameters. The principal solution to this minimax problem is presented and the resulting decision rule is discussed. Since the exact solution is, in general, hard to find, and afortiori hard to implement, an approximation method that yields an asymptotically minimax decision rule is proposed. Finally, a variety of potential application areas are provided in signal processing and communications with special emphasis on universal decoding.
Relations between random coding exponents and the statistical physics of random codes
 IEEE Trans. Inf. Theory
, 2009
"... The partition function pertaining to finite–temperature decoding of a (typical) randomly chosen code is known to have three types of behavior, corresponding to three phases in the plane of rate vs. temperature: the ferromagnetic phase, corresponding to correct decoding, the paramagnetic phase, of co ..."
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Cited by 15 (14 self)
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The partition function pertaining to finite–temperature decoding of a (typical) randomly chosen code is known to have three types of behavior, corresponding to three phases in the plane of rate vs. temperature: the ferromagnetic phase, corresponding to correct decoding, the paramagnetic phase, of complete disorder, which is dominated by exponentially many incorrect codewords, and the glassy phase (or the condensed phase), where the system is frozen at minimum energy and dominated by subexponentially many incorrect codewords. We show that the statistical physics associated with the two latter phases are intimately related to random coding exponents. In particular, the exponent associated with the probability of correct decoding at rates above capacity is directly related to the free energy in the glassy phase, and the exponent associated with probability of error (the error exponent) at rates below capacity, is strongly related to the free energy in the paramagnetic phase. In fact, we derive alternative expressions of these exponents in terms of the corresponding free energies, and make an attempt to obtain some insights from these expressions. Finally, as a side result, we also compare the phase diagram associated with a simple finite–temperature universal decoder, for discrete memoryless channels, to that of the finite–temperature decoder that is aware of the channel statistics. Index Terms: random coding, free energy, partition function, random energy model (REM), phase transitions, error exponents.
Achievable error exponents for the private fingerprinting game
 IEEE Trans. Information Theory
, 2007
"... Fingerprinting systems in the presence of collusive attacks are analyzed as a game between a fingerprinter and a decoder, on the one hand, and a coalition of two or more attackers, on the other hand. The fingerprinter distributes, to different users, different fingerprinted copies of a host data (co ..."
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Cited by 11 (4 self)
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Fingerprinting systems in the presence of collusive attacks are analyzed as a game between a fingerprinter and a decoder, on the one hand, and a coalition of two or more attackers, on the other hand. The fingerprinter distributes, to different users, different fingerprinted copies of a host data (covertext), drawn from a memoryless stationary source, embedded with different fingerprints. The coalition members create a forgery of the data while aiming at erasing the fingerprints in order not to be detected. Their action is modelled by a multiple access channel (MAC). We analyze the performance of two classes of decoders, associated with different kinds of error events. The decoder of the first class aims at detecting the entire coalition, whereas the second is satisfied with the detection of at least one member of the coalition. Both decoders have access to the original covertext data and observe the forgery in order to identify member/s of the coalition. Motivated by a worstcase approach, we assume that the coalition of attackers is informed of the hiding strategy taken by the fingerprinter and the decoder, while they are uninformed of the attacking scheme. Single letter expressions for the error exponents of the two kinds are obtained, a decoder that is optimal with respect to the two kinds of errors is introduced, and the worstcase attack channel is characterized. 1
The Generalized Random Energy Model and its Application to the Statistical Physics of Ensembles of Hierarchical Codes
, 2007
"... In an earlier work, the statistical physics associated with finite–temperature decoding of code ensembles, along with the relation to their random coding error exponents, were explored in a framework that is analogous to Derrida’s random energy model (REM) of spin glasses, according to which the ene ..."
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Cited by 8 (8 self)
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In an earlier work, the statistical physics associated with finite–temperature decoding of code ensembles, along with the relation to their random coding error exponents, were explored in a framework that is analogous to Derrida’s random energy model (REM) of spin glasses, according to which the energy levels of the various spin configurations are independent random variables. The generalized REM (GREM) extends the REM in that it introduces correlations between energy levels in an hierarchical structure. In this paper, we explore some analogies between the behavior of the GREM and that of code ensembles which have parallel hierarchical structures. In particular, in analogy to the fact that the GREM may have different types of phase transition effects, depending on the parameters of the model, then the above–mentioned hierarchical code ensembles behave substantially differently in the various domains of the design parameters of these codes. We make an attempt to explore the insights that can be imported from the statistical mechanics of the GREM and be harnessed to serve for code design considerations and guidelines.
Lower bounds on the error probability of block codes based on improvements on de Caen’s inequality
 IEEE TRANS. INFORM. THEORY
, 2004
"... New lower bounds on the error probability of block codes with maximumlikelihood decoding are proposed. The bounds are obtained by applying a new lower bound on the probability of a union of events, derived by improving on de Caen’s lower bound. The new bound includes an arbitrary function to be op ..."
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Cited by 7 (0 self)
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New lower bounds on the error probability of block codes with maximumlikelihood decoding are proposed. The bounds are obtained by applying a new lower bound on the probability of a union of events, derived by improving on de Caen’s lower bound. The new bound includes an arbitrary function to be optimized in order to achieve the tightest results. Since the optimal choice of this function is known, but leads to a trivial and useless identity, we find several useful approximations for it, each resulting in a new lower bound. For the additive white Gaussian noise (AWGN) channel and the binarysymmetric channel (BSC), the optimal choice of the optimization function is stated and several approximations are proposed. When the bounds are further specialized to linear codes, the only knowledge on the code used is its weight enumeration. The results are shown to be tighter than the latest bounds in the current literature, such as those by Seguin and by Keren and Litsyn. Moreover, for the BSC, the new bounds widen the range of rates for which the union bound analysis applies, thus improving on the bound to the error exponent compared with the de Caenbased bounds.
On Random Coding Error Exponents of Watermarking Codes
 IEEE Trans. Info Thy
, 2000
"... steganography, watermarking, information hiding, error exponent, random coding Watermarking codes are analyzed from an informationtheoretic viewpoint as a game between an information hider and an active attacker. While the information hider embeds a secret message (watermark) in a covertext message ..."
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Cited by 6 (2 self)
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steganography, watermarking, information hiding, error exponent, random coding Watermarking codes are analyzed from an informationtheoretic viewpoint as a game between an information hider and an active attacker. While the information hider embeds a secret message (watermark) in a covertext message (typically, text, image, sound, or video stream) within a certain distortion level, the attacker processes the resulting watermarked message, within limited additional distortion, in an attempt to invalidate the watermark. For a memoryless covertext source, we provide a singleletter characterization of the minimaxmaximin game of the random coding error exponent associated with the average probability of erroneously decoding the watermark. This singleletter characterization is in effect because there is a “memoryless saddle point ” in this game: The information hider utilizes a memoryless channel to generate random codewords for every covertext message, whereas the attacker implements a memoryless channel to disrupt the watermark information hidden in the covertext.
Coding over an Erasure Channel with a Large Alphabet Size
, 2008
"... An erasure channel with a fixed alphabet size q, where q ≫ 1, is studied. It is proved that over any erasure channel (with or without memory), Maximum Distance Separable (MDS) codes achieve the minimum probability of error (assuming maximum likelihood decoding). Assuming a memoryless erasure channel ..."
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Cited by 5 (2 self)
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An erasure channel with a fixed alphabet size q, where q ≫ 1, is studied. It is proved that over any erasure channel (with or without memory), Maximum Distance Separable (MDS) codes achieve the minimum probability of error (assuming maximum likelihood decoding). Assuming a memoryless erasure channel, the error exponent of MDS codes are compared with that of random codes. It is shown that the envelopes of these two exponents are identical for rates above the critical rate. Noting the optimality of MDS codes, it is concluded that random coding is exponentially optimal as long as the block size N satisfies N < q + 1.
A new universal randomcoding bound for average probability error exponent for multipleaccess channels
 IN PROC. CONFERENCE ON INFORMATION SCIENCES AND SYSTEMS (CISS), MAR. 2009, ONLINE: HTTP://ARXIV.ORG/ABS/0901.0948
, 2009
"... In this work, a new upper bound for average error probability of a twouser discrete memoryless (DM) multipleaccess channel (MAC) is derived. This bound can be universally obtained for all discrete memoryless MACs with given input and output alphabets. This is the first bound of this type that expl ..."
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Cited by 3 (3 self)
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In this work, a new upper bound for average error probability of a twouser discrete memoryless (DM) multipleaccess channel (MAC) is derived. This bound can be universally obtained for all discrete memoryless MACs with given input and output alphabets. This is the first bound of this type that explicitly uses the method of expurgation. It is shown that the exponent of this bound is greater than or equal to those of previously known bounds.
Error Exponent for MultipleAccess Channels: Lower Bounds
, 2010
"... A unified framework to obtain all known lower bounds (random coding, typical random coding and expurgated bound) on the reliability function of a pointtopoint discrete memoryless channel (DMC) is presented. By using a similar idea for a twouser discrete memoryless (DM) multipleaccess channel (MA ..."
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Cited by 1 (1 self)
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A unified framework to obtain all known lower bounds (random coding, typical random coding and expurgated bound) on the reliability function of a pointtopoint discrete memoryless channel (DMC) is presented. By using a similar idea for a twouser discrete memoryless (DM) multipleaccess channel (MAC), three lower bounds on the reliability function are derived. The first one (random coding) is identical to the best known lower bound on the reliability function of DMMAC. It is shown that the random coding bound is the performance of the average code in the constant composition code ensemble. The second bound (Typical random coding) is the typical performance of the constant composition code ensemble. To derive the third bound (expurgated), we eliminate some of the codewords from the codebook with larger rate. This is the first bound of this type that explicitly uses the method of expurgation for MACs. It is shown that the exponent of the typical random coding and the expurgated bounds are greater than or equal to the exponent of the known random coding bounds for all rate pairs. Moreover, an example is given where the exponent of the expurgated bound is strictly larger. All these bounds can be universally obtained for all discrete memoryless MACs with given input and output alphabets.