Searcharxiv⌕ Search

arXiv subjects

Marco Dalai

Publications and source records attributed to Marco Dalai.

35 records · Page 2Linked to original sources
arXiv2019

A gap in the slice rank of $k$-tensors

The slice-rank method, introduced by Tao as a symmetrized version of the polynomial method of Croot, Lev and Pach and Ellenberg and Gijswijt, has proved to be a useful tool in a variety of combinatorial problems. Explicit tensors have been introduced in different contexts but little is known about the limitations of the method. In this paper, building upon a method presented by Tao and Sawin, it is proved that the asymptotic slice rank of any $k$-tensor in any field is either $1$ or at least $k/(k-1)^{(k-1)/k}$. This provides evidence that straight-forward application of the method cannot give useful results in certain problems for which non-trivial exponential bounds are already known. An example, actually a motivation for starting this work, is the problem of bounding the size of trifferent sets of sequences, which constitutes a long-standing open problem in information theory and in theoretical computer science.

math.CO↗
arXiv2016

Nonasymptotic coding-rate bounds for binary erasure channels with feedback

We present nonasymptotic achievability and converse bounds on the maximum coding rate (for a fixed average error probability and a fixed average blocklength) of variable-length full-feedback (VLF) and variable-length stop-feedback (VLSF) codes operating over a binary erasure channel (BEC). For the VLF setup, the achievability bound relies on a scheme that maps each message onto a variable-length Huffman codeword and then repeats each bit of the codeword until it is received correctly. The converse bound is inspired by the meta-converse framework by Polyanskiy, Poor, and Verdú (2010) and relies on binary sequential hypothesis testing. For the case of zero error probability, our achievability and converse bounds match. For the VLSF case, we provide achievability bounds that exploit the following feature of BEC: the decoder can assess the correctness of its estimate by verifying whether the chosen codeword is the only one that is compatible with the erasure pattern. One of these bounds is obtained by analyzing the performance of a variable-length extension of random linear fountain codes. The gap between the VLSF achievability and the VLF converse bound, when number of messages is small, is significant: $23\%$ for 8 messages on a BEC with erasure probability $0.5.$ The absence of a tight VLSF converse bound does not allow us to assess whether this gap is fundamental.

cs.IT↗
arXiv2016

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 $δ$ 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 $δ$ 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 Lovász' theta function with the construction used by McEliece et al. for bounding the minimum distance of codes in Hamming spaces.

cs.IT↗
arXiv2016

Bounds on the Reliability of a Typewriter Channel

We give new bounds on the reliability function of a typewriter channel with 5 inputs and crossover probability $1/2$. The lower bound is more of theoretical than practical importance; it improves very marginally the expurgated bound, providing a counterexample to a conjecture on its tightness by Shannon, Gallager and Berlekamp which does not need the construction of algebraic-geometric codes previously used by Katsman, Tsfasman and Vlăduţ. The upper bound is derived by using an adaptation of the linear programming bound and it is essentially useful as a low-rate anchor for the straight line bound.

cs.IT↗
arXiv2015

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 Lovász 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 Lovász 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.

cs.IT↗
arXiv2015

Bounds for codes on pentagon and other cycles

The capacity of a graph is defined as the rate of exponential grow of independent sets in the strong powers of the graph. In strong power, an edge connects two sequences if at each position letters are equal or adjacent. We consider a variation of the problem where edges in the power graphs are removed among sequences which differ in more than a fraction $δ$ of coordinates. For odd cycles, we derive an upper bound on the corresponding rate which combines Lovász' bound on the capacity with Delsarte's linear programming bounds on the minimum distance of codes in Hamming spaces. For the pentagon, this shows that for $δ\ge {1-{1\over\sqrt{5}}}$ the Lovász rate is the best possible, while we prove by a Gilbert-Varshamov-type bound that a higher rate is achievable for $δ< {2\over 5}$. Communication interpretation of this question is the problem of sending quinary symbols subject to $\pm 1\mod 5$ disturbance. The maximal communication rate subject to the zero undetected-error equals capacity of a pentagon. The question addressed here is how much this rate can be increased if only a fraction $δ$ of symbols is allowed to be disturbed

math.CO↗
arXiv2015

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 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.

cs.IT↗
arXiv2015

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, Lovász and Marton. The result can in fact be interpreted as a unification of the Elias bound and of Lovász'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.

cs.IT↗
arXiv2014

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 Lovász 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 Lovász 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.

cs.IT↗
arXiv2013

Lovász's Theta Function, Rényi's Divergence and the Sphere-Packing Bound

Lovász'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 Csiszár type using the R{é}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.

cs.IT↗
arXiv2013

An "Umbrella" Bound of the Lovász-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 Lovász' and Gallager's ideas. The obtained bounds are expressed in terms of a function $\vartheta(ρ)$, introduced here, that varies from the cut-off rate of the channel to the Lovázs theta function as $ρ$ varies from 1 to $\infty$ 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 Lovász theta function.

cs.IT↗
arXiv2012

How would Riemann evaluate $ζ(2n)$?

Commenting on an observation of Prof. Edwards, this note presents a method of evaluation of $ζ(2n)$ that follows easily from Riemann's own representation of the zeta function.

math.HO↗
arXiv2012

A new bound for the capacity of the deletion channel with high deletion probabilities

Let $C(d)$ be the capacity of the binary deletion channel with deletion probability $d$. It was proved by Drinea and Mitzenmacher that, for all $d$, $C(d)/(1-d)\geq 0.1185 $. Fertonani and Duman recently showed that $\limsup_{d\to 1}C(d)/(1-d)\leq 0.49$. In this paper, it is proved that $\lim_{d\to 1}C(d)/(1-d)$ exists and is equal to $\inf_{d}C(d)/(1-d)$. This result suggests the conjecture that the curve $C(d)$ my be convex in the interval $d\in [0,1]$. Furthermore, using currently known bounds for $C(d)$, it leads to the upper bound $\lim_{d\to 1}C(d)/(1-d)\leq 0.4143$.

cs.IT↗
arXiv2008

On Unique Decodability

In this paper we propose a revisitation of the topic of unique decodability and of some fundamental theorems of lossless coding. It is widely believed that, for any discrete source X, every "uniquely decodable" block code satisfies E[l(X_1 X_2 ... X_n)]>= H(X_1,X_2,...,X_n), where X_1, X_2,...,X_n are the first n symbols of the source, E[l(X_1 X_2 ... X_n)] is the expected length of the code for those symbols and H(X_1,X_2,...,X_n) is their joint entropy. We show that, for certain sources with memory, the above inequality only holds when a limiting definition of "uniquely decodable code" is considered. In particular, the above inequality is usually assumed to hold for any "practical code" due to a debatable application of McMillan's theorem to sources with memory. We thus propose a clarification of the topic, also providing an extended version of McMillan's theorem to be used for Markovian sources.

cs.IT↗
arXiv2005

Non prefix-free codes for constrained sequences

In this paper we consider the use of variable length non prefix-free codes for coding constrained sequences of symbols. We suppose to have a Markov source where some state transitions are impossible, i.e. the stochastic matrix associated with the Markov chain has some null entries. We show that classic Kraft inequality is not a necessary condition, in general, for unique decodability under the above hypothesis and we propose a relaxed necessary inequality condition. This allows, in some cases, the use of non prefix-free codes that can give very good performance, both in terms of compression and computational efficiency. Some considerations are made on the relation between the proposed approach and other existing coding paradigms.

cs.IT↗
arXiv2004

Recurrence relations for the Lerch Phi function and applications

In this paper we present a simple method for deriving recurrence relations and we apply it to obtain two equations involving the Lerch Phi function and sums of Bernoulli and Euler polynomials. Connections between these results and those obtained in a paper of H.M. Srivastava, M.L. Glasser and V. Adamchik are pointed out, emphasizing the usefulness of this approach with some meaningful examples.

math.NT↗