1,721,017 research outputs found
An Elias bound on the Bhattacharyya distance of codes for channels with a zero-error capacity
No full textIn this paper, we propose an upper bound on the minimum Bhattacharyya distance of codes for channels with a zero-error capacity. The bound is obtained by combining an extension of the Elias bound introduced by Blahut, with an extension of a bound previously introduced by the author, which builds upon ideas of Gallager, Lovász and Marton
How Would Riemann Evaluate ζ(2n)?
No full textDriven by an inspiring comment by Prof. H. M. Edwards, we present a method of evaluation of zeta(2n), apparently unnoticed before, that follows easily from Riemann's own representation of the zeta function
Lower Bounds on the Probability of Error for Classical and Classical-Quantum Channels
No full textIn this paper, lower bounds on error probability in coding for discrete classical and classical-quantum channels are studied. The contribution of the paper goes in two main directions: 1) extending classical bounds of Shannon to classical-quantum channels, and 2) proposing a new framework for lower bounding the probability of error of channels with a zero-error capacity in the low rate region. The relation between these two problems is revealed by showing that Lovász' bound on zero-error capacity emerges as a natural consequence of the sphere packing bound once we move to the more general context of classical-quantum channels. A variation of Lovász' bound is then derived to lower bound the probability of error in the low rate region by means of auxiliary channels. As a result of this study, connections between the Lovász theta function, the expurgated bound of Gallager, the cutoff rate of a classical channel, and the sphere packing bound for classical-quantum channels are established
Some Remarks on Classical and Classical-Quantum Sphere Packing Bounds: Rényi vs. Kullback–Leibler
No full textWe review the use of binary hypothesis testing for the derivation of the sphere packing bound in channel coding, pointing out a key difference between the classical and the classical-quantum setting. In the first case, two ways of using the binary hypothesis testing are known, which lead to the same bound written in different analytical expressions. The first method historically compares output distributions induced by the codewords with an auxiliary fixed output distribution, and naturally leads to an expression using the Renyi divergence. The second method compares the given channel with an auxiliary one and leads to an expression using the Kullback–Leibler divergence. In the classical-quantum case, due to a fundamental difference in the quantum binary hypothesis testing, these two approaches lead to two different bounds, the first being the “right” one. We discuss the details of this phenomenon, which suggests the question of whether auxiliary channels are used in the optimal way in the second approach and whether recent results on the exact strong-converse exponent in classical-quantum channel coding might play a role in the considered problem
Sphere packing bound for quantum channels
No full textIn this paper, the Sphere-Packing-Bound of Fano, Shannon, Gallager and Berlekamp is extended to general classical-quantum channels. The obtained upper bound for the reliability function, for the case of pure-state channels, coincides at high rates with a lower bound derived by Burnashev and Holevo [1]. Thus, for pure state channels, the reliability function at high rates is now exactly determined. For the general case, the obtained upper bound expression at high rates was conjectured to represent also a lower bound to the reliability function, but a complete proof has not been obtained yet
A New Bound on the Capacity of the Binary Deletion Channel with High Deletion Probabilities
No full textLet be the capacity of the binary deletion channel with deletion probability . It was proved by
Drinea and Mitzenmacher that, for all ,
. Fertonani and Duman recently showed that . In this paper, it is proved that
exists and is equal to . This result suggests the conjecture that the curve my be convex in the interval . Furthermore,
using currently known bounds for , it leads to the upper bound
Elias Bound for General Distances and Stable Sets in Edge-Weighted Graphs
No full textThis paper presents an extension of the Elias bound on the minimum distance of codes for discrete alphabets with general, possibly infinite-valued, distances. The bound is obtained by combining a previous extension of the Elias bound, introduced by Blahut, with an extension of a bound previously introduced by the author which builds upon ideas of Gallager, Lova ́sz and Marton. The result can in fact be interpreted as a unification of the Elias bound and of Lova ́sz’s bound on graph (or zero-error) capacity, both being recovered as particular cases of the one presented here. Previous extensions of the Elias bound by Berlekamp, Blahut and Piret are shown to be included as particular cases of our bound. Applications to the reliability function are then discussed
Minimal Information Exchange for Image Registration
No full textIn this paper we consider the problem of estimating the relative shift, scale and rotation between two images X and Y that are available to two users, respectively A and B, connected through a channel. User A is asked to send B some specifically selected minimal description of image X that will allow B to recover the relative shift, rotation and scale between X and Y. The approach is based on a distributed encoding technique applied to the Discrete Fourier Transform phase and to the Fourier-Mellin transform of the images
L-infinity norm based second generation image coding
No full textMany second generation image coding techniques have been studied in recent years. Most of these methods consider the l2 norm of the error introduced in the coded image, while for the l∞ case only predictive or transform based methods were considered up to now, focusing on near-lossless coding. In this paper we present a first scheme for l∞ norm in the framework of second generation image coding. The image is adaptively segmented into rectangular regions of varying size leading to a binary tree decomposition. The grey levels of the pixels within every leaf are approximated by means of l∞ sub-optimal bilinear surfaces
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