83 research outputs found
An "Umbrella" Bound of the Lov\'asz-Gallager Type
We propose a novel approach for bounding the probability of error of discrete
memoryless channels with a zero-error capacity based on a combination of
Lov\'asz' and Gallager's ideas. The obtained bounds are expressed in terms of a
function , introduced here, that varies from the cut-off rate
of the channel to the Lov\'azs theta function as varies from 1 to
and which is intimately related to Gallager's expurgated coefficient.
The obtained bound to the reliability function, though loose in its present
form, is finite for all rates larger than the Lov\'asz theta function.Comment: An excerpt from arXiv:1201.5411v3 (with a classical notation)
accepted at ISIT 201
Elias Bound for General Distances and Stable Sets in Edge-Weighted Graphs
This 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, Lov\'asz and Marton. The
result can in fact be interpreted as a unification of the Elias bound and of
Lov\'asz'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.Comment: Accepted, IEEE Transaction on Information Theor
Lov\'asz's Theta Function, R\'enyi's Divergence and the Sphere-Packing Bound
Lov\'asz's bound to the capacity of a graph and the the sphere-packing bound
to the probability of error in channel coding are given a unified presentation
as information radii of the Csisz\'ar type using the R{\'e}nyi divergence in
the classical-quantum setting. This brings together two results in coding
theory that are usually considered as being of a very different nature, one
being a "combinatorial" result and the other being "probabilistic". In the
context of quantum information theory, this difference disappears.Comment: An excerpt from arXiv:1201.5411v3 (with a different notation)
accepted at ISIT 201
Constant Compositions in the Sphere Packing Bound for Classical-Quantum Channels
The sphere packing bound, in the form given by Shannon, Gallager and
Berlekamp, was recently extended to classical-quantum channels, and it was
shown that this creates a natural setting for combining probabilistic
approaches with some combinatorial ones such as the Lov\'asz theta function. In
this paper, we extend the study to the case of constant composition codes. We
first extend the sphere packing bound for classical-quantum channels to this
case, and we then show that the obtained result is related to a variation of
the Lov\'asz theta function studied by Marton. We then propose a further
extension to the case of varying channels and codewords with a constant
conditional composition given a particular sequence. This extension is then
applied to auxiliary channels to deduce a bound which can be interpreted as an
extension of the Elias bound.Comment: ISIT 2014. Two issues that were left open in Section IV of the first
version are now solve
Some remarks on classical and classical-quantum sphere packing bounds: Rényi vs. Kullback-Leibler
We 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 proble
Rate-distance tradeoff for codes above graph capacity
The capacity of a graph is defined as the rate of exponential growth of
independent sets in the strong powers of the graph. In the strong power an edge
connects two sequences if at each position their letters are equal or adjacent.
We consider a variation of the problem where edges in the power graphs are
removed between sequences which differ in more than a fraction of
coordinates. The proposed generalization can be interpreted as the problem of
determining the highest rate of zero undetected-error communication over a link
with adversarial noise, where only a fraction of symbols can be
perturbed and only some substitutions are allowed.
We derive lower bounds on achievable rates by combining graph homomorphisms
with a graph-theoretic generalization of the Gilbert-Varshamov bound. We then
give an upper bound, based on Delsarte's linear programming approach, which
combines Lov\'asz' theta function with the construction used by McEliece et al.
for bounding the minimum distance of codes in Hamming spaces.Comment: 5 pages. Presented at 2016 IEEE International Symposium on
Information Theor
A New Bound on the Capacity of the Binary Deletion Channel with High Deletion Probabilities
Let 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
Lower bounds on the Probability of Error for Classical and Classical-Quantum Channels
In 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: i) extending classical bounds of Shannon,
Gallager and Berlekamp to classical-quantum channels, and ii) 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 Lov\'asz' 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
Lov\'asz' 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 Lov\'asz 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.Comment: Updated to published version + bug fixed in Figure
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