37 research outputs found
Coding against synchronisation and related errors
In this thesis, we study aspects of coding against synchronisation errors, such as deletions and replications, and related errors. Synchronisation errors are a source of fundamental open problems in information theory, because they introduce correlations between output symbols even when input symbols are independently distributed. We focus on random errors, and consider two complementary problems:
We study the optimal rate of reliable information transmission through channels with synchronisation and related errors (the channel capacity). Unlike simpler error models, the capacity of such channels is unknown. We first consider the geometric sticky channel, which replicates input bits according to a geometric distribution. Previously, bounds on its capacity were known only via numerical methods, which do not aid our conceptual understanding of this quantity. We derive sharp analytical capacity upper bounds which approach, and sometimes surpass, numerical bounds. This opens the door to a mathematical treatment of its capacity. We consider also the geometric deletion channel, combining deletions and geometric replications. We derive analytical capacity upper bounds, and notably prove that the capacity is bounded away from the maximum when the deletion probability is small, meaning that this channel behaves differently than related well-studied channels in this regime. Finally, we adapt techniques developed to handle synchronisation errors to derive improved upper bounds and structural results on the capacity of the discrete-time Poisson channel, a model of optical communication.
Motivated by portable DNA-based storage and trace reconstruction, we introduce and study the coded trace reconstruction problem, where the goal is to design efficiently encodable high-rate codes whose codewords can be efficiently reconstructed from few reads corrupted by deletions. Remarkably, we design such n-bit codes with rate 1-O(1/log n) that require exponentially fewer reads than average-case trace reconstruction algorithms.Open Acces
Sharp Bounds on the Entropy of the Poisson Law and Related Quantities
One of the difficulties in calculating the capacity of certain Poisson
channels is that H(lambda), the entropy of the Poisson distribution with mean
lambda, is not available in a simple form. In this work we derive upper and
lower bounds for H(lambda) that are asymptotically tight and easy to compute.
The derivation of such bounds involves only simple probabilistic and analytic
tools. This complements the asymptotic expansions of Knessl (1998), Jacquet and
Szpankowski (1999), and Flajolet (1999). The same method yields tight bounds on
the relative entropy D(n, p) between a binomial and a Poisson, thus refining
the work of Harremoes and Ruzankin (2004). Bounds on the entropy of the
binomial also follow easily.Comment: To appear, IEEE Trans. Inform. Theor
A refined analysis of the Poisson channel in the high-photon-efficiency regime
We study the discrete-time Poisson channel under the constraint that its
average input power (in photons per channel use) must not exceed some constant
E. We consider the wideband, high-photon-efficiency extreme where E approaches
zero, and where the channel's "dark current" approaches zero proportionally
with E. Improving over a previously obtained first-order capacity
approximation, we derive a refined approximation, which includes the exact
characterization of the second-order term, as well as an asymptotic
characterization of the third-order term with respect to the dark current. We
also show that pulse-position modulation is nearly optimal in this regime.Comment: Revised version to appear in IEEE Transactions on Information Theor
Poisson noise channel with dark current: Numerical computation of the optimal input distribution
This paper considers a discrete time-Poisson noise channel which is used to
model pulse-amplitude modulated optical communication with a direct-detection
receiver. The goal of this paper is to obtain insights into the capacity and
the structure of the capacity-achieving distribution for the channel under the
amplitude constraint and in the presence of dark current
. Using recent theoretical progress on the structure of the
capacity-achieving distribution, this paper develops a numerical algorithm,
based on the gradient ascent and Blahut-Arimoto algorithms, for computing the
capacity and the capacity-achieving distribution. The algorithm is used to
perform extensive numerical simulations for various regimes of and
.Comment: Submitted to IEEE ICC 2022. This is a companion paper of: A. Dytso,
L. Barletta and S. Shamai Shitz, "Properties of the Support of the
Capacity-Achieving Distribution of the Amplitude-Constrained Poisson Noise
Channel," in IEEE Transactions on Information Theory, vol. 67, no. 11, pp.
7050-7066, Nov. 202