Lower bounds to error probability for coding on discrete memoryless channels. II

Shannon, Claude E.; Gallager, Robert G.; Berlekamp, Elwyn R.

doi:10.1016/s0019-9958(67)91200-4

Cited by 266 publications

(257 citation statements)

References 6 publications

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“…Thus, by Theorem 7, for these values of we have . The full claim follows from the straight-line bound of Shannon, Gallager, and Berlekamp [27]. Remark: We have seen in Lemma 4 that for , it suffices to rely on the simple form of the function , namely, .…”

Section: Theorem 17mentioning

confidence: 96%

“…Several important ideas in this problem were suggested in the paper [27]. The nature of the upper bounds is different for low values of and for close to capacity.…”

Section: A Error Exponentsmentioning

confidence: 99%

“…The function is the linear programming bound of [23] . Summing both sides of the last inequality on from to , we obtain the estimate of in the form (4) Recall from [27] that a straight-line segment that connects a point on with a point on any other upper bound on is also a valid upper bound on . This result is called the straight-line principle.…”

Section: Theoremmentioning

confidence: 99%

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Distance Distribution of Binary Codes and the Error Probability of Decoding

Barg

McGregor²

2005

IEEE Trans. Inform. Theory

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Abstract-We address the problem of bounding below the probability of error under maximum-likelihood decoding of a binary code with a known distance distribution used on a binarysymmetric channel (BSC). An improved upper bound is given for the maximum attainable exponent of this probability (the reliability function of the channel). In particular, we prove that the "random coding exponent" is the true value of the channel reliability for codes rate in some interval immediately below the critical rate of the channel. An analogous result is obtained for the Gaussian channel.

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Section: Theorem 17mentioning

confidence: 96%

“…Several important ideas in this problem were suggested in the paper [27]. The nature of the upper bounds is different for low values of and for close to capacity.…”

Section: A Error Exponentsmentioning

confidence: 99%

Section: Theoremmentioning

confidence: 99%

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Distance Distribution of Binary Codes and the Error Probability of Decoding

Barg

McGregor²

2005

IEEE Trans. Inform. Theory

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“…VI.E] to be equal to the sphere-packing exponent. However, below the critical rate of the channel, the reliability function is in general bounded away [7] from the sphere-packing exponent and thus, the gap between (52) and (53) may grow exponentially with n. 1 shows different bounds on the error probability for M = 4 messages and a binary symmetric channel (BSC). In this setup, the best code can be obtained explicitly [8] and hence the exact ML decoding error probability can be computed.…”

Section: Connection With Previous Resultsmentioning

confidence: 99%

The meta-converse bound is tight

Vazquez-Vilar

Campo

Fàbregas

et al. 2013

2013 IEEE International Symposium on Information Theory

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“…Such would be the case, for instance, of random expurgated codes [16,17] and random constant-composition codes [18].…”

Section: Mathematical Modelmentioning

confidence: 99%

Block QIM Watermarking Games

Moulin

Goteti

2006

IEEE Trans.Inform.Forensic Secur.

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While binning is a fundamental approach to blind data embedding and watermarking, an attacker may devise various strategies to reduce the effectiveness of practical binning schemes. The problem analyzed in this paper is design of worst-case noise distributions against L-dimensional lattice Quantization Index Modulation (QIM) watermarking codes. The cost functions considered are (1) probability of error of the maximum-likelihood decoder, and (2) the more tractable Bhattacharyya upper bound on error probability, which is tight at low embedding rates. Both problems are addressed under the following constraints on the attacker's strategy: the noise is independent of the marked signal, blockwise memoryless with block length L, and may not exceed a specified quadratic-distortion level. The embedder's quadratic distortion is limited as well. Three strategies are considered for the embedder: optimization of the lattice inflation parameter (aka Costa parameter), dithering, and randomized lattice rotation. Critical in this analysis are the symmetry properties of QIM nested lattices and convexity properties of probability of error and related functionals of the noise distribution. We derive the minmax optimal embedding and attack strategies and obtain explicit solutions as well as numerical solutions for the worst-case noise. The role of the attacker's memory is investigated; in particular, we demonstrate the remarkable effectiveness of impulsive-noise attacks as L increases. The formulation proposed in this paper is also used to evaluate the capacity of lattice QIM under worst-noise conditions.

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Lower bounds to error probability for coding on discrete memoryless channels. II

Cited by 266 publications

References 6 publications

Distance Distribution of Binary Codes and the Error Probability of Decoding

Distance Distribution of Binary Codes and the Error Probability of Decoding

The meta-converse bound is tight

Block QIM Watermarking Games

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