Computer Science

This paper investigates total variation minimization in one spatial dimension for the recovery of gradient-sparse signals from undersampled Gaussian measurements. Recently established bounds for the required sampling rate state that uniform recovery of all s -gradient-sparse signals in R n is only possible with m≳ sn − − √ ⋅PolyLog(n) measurements. Such a condition is especially prohibitive for high-dimensional problems, where s is much smaller than n . However, previous empirical findings seem to indicate that the latter sampling rate does not reflect the typical behavior of total variation minimization. Indeed, this work provides a rigorous analysis that breaks the sn − − √ -bottleneck for a large class of natural signals. The main result shows that non-uniform recovery succeeds with high probability for m≳s⋅PolyLog(n) measurements if the jump discontinuities of the signal vector are sufficiently well separated. In particular, this guarantee allows for signals arising from a discretization of piecewise constant functions defined on an interval. The present paper serves as a short summary of the main results in our recent work [arXiv:2001.09952].

Probabilistic shaping (PS) has been widely studied and applied to optical fiber communications. The encoder of PS expends the number of bit slots and controls the probability distribution of channel input symbols. Not only studies focused on PS but also most works on optical fiber communications have assumed source uniformity (i.e. equal probability of marks and spaces) so far. On the other hand, the source information is in general nonuniform, unless bit-scrambling or other source coding techniques to balance the bit probability is performed. Interestingly, one can exploit the source nonuniformity to reduce the entropy of the channel input symbols with the PS encoder, which leads to smaller required signal-to-noise ratio at a given input logic rate. This benefit is equivalent to a combination of data compression and PS, and thus we call this technique compressed shaping. In this work, we explain its theoretical background in detail, and verify the concept by both numerical simulation and a field programmable gate array (FPGA) implementation of such a system. In particular, we find that compressed shaping can reduce power consumption in forward error correction decoding by up to 90% in nonuniform source cases. The additional hardware resources required for compressed shaping are not significant compared with forward error correction coding, and an error insertion test is successfully demonstrated with the FPGA.

The channel reliability function is an important tool that characterizes the reliable transmission of messages over communications channels. For many channels only the upper and lower bounds of the function are known. In this paper we analyze the computability of the reliability function and its related functions. We show that the reliability function is not a Turing computable performance function. The same also applies to the functions of the sphere packing bound and the expurgation bound. Furthermore, we consider the R ??function and the zero-error feedback capacity, both of them play an important role for the reliability function. Both the R ??function and the zero-error feedback capacity are not Banach Mazur computable. We show that the R ??function is additive. The zero-error feedback capacity is super-additive and we characterize it's behavior.

We consider reliable and secure communication over intersymbol interference wiretap channels (ISI-WTCs). In particular, we first examine the setup where the source at the input of an ISI-WTC is unconstrained and then, based on a general achievability result for arbitrary wiretap channels, we derive an achievable secure information rate for this ISI-WTC. Afterwards, we examine the setup where the source at the input of an ISI-WTC is constrained to be a finite-state machine source (FSMS) of a certain order and structure. Optimizing the parameters of this FSMS toward maximizing the secure information rate is a computationally intractable problem in general, and so, toward finding a local maximum, we propose an iterative algorithm that at every iteration replaces the secure information rate function by a suitable surrogate function whose maximum can be found efficiently. Although the secure information rates achieved in the unconstrained setup are expected to be larger than the secure information rates achieved in the constrained setup, the latter setup has the advantage of leading to efficient algorithms for estimating achievable secure rates and also has the benefit of being the basis of efficient encoding and decoding schemes.

In 2020, Budaghyan, Helleseth and Kaleyski [IEEE TIT 66(11): 7081-7087, 2020] considered an infinite family of quadrinomials over F 2 n of the form x 3 +a( x 2 s +1 ) 2 k +b x 3??2 m +c( x 2 s+m + 2 m ) 2 k , where n=2m with m odd. They proved that such kind of quadrinomials can provide new almost perfect nonlinear (APN) functions when gcd(3,m)=1 , k=0 , and (s,a,b,c)=(m??,?, ? 2 ,1) or ((m?? ) ?? mod n,?, ? 2 ,1) in which ???F 4 ??F 2 . By taking a=? and b=c= ? 2 , we observe that such kind of quadrinomials can be rewritten as a Tr n m (b x 3 )+ a q Tr n m (c x 2 s +1 ) , where q= 2 m and Tr n m (x)=x+ x 2 m for n=2m . Inspired by the quadrinomials and our observation, in this paper we study a class of functions with the form f(x)=a Tr n m (F(x))+ a q Tr n m (G(x)) and determine the APN-ness of this new kind of functions, where a??F 2 n such that a+ a q ?? , and both F and G are quadratic functions over F 2 n . We first obtain a characterization of the conditions for f(x) such that f(x) is an APN function. With the help of this characterization, we obtain an infinite family of APN functions for n=2m with m being an odd positive integer: f(x)=a Tr n m (b x 3 )+ a q Tr n m ( b 3 x 9 ) , where a??F 2 n such that a+ a q ?? and b is a non-cube in F 2 n .

We give two methods for constructing many linear complementary dual (LCD for short) codes from a given LCD code, by modifying some known methods for constructing self-dual codes. Using the methods, we construct binary LCD codes and quaternary Hermitian LCD codes, which improve the previously known lower bound on the largest minimum weights.

Optimal design of distributed decision policies can be a difficult task, illustrated by the famous Witsenhausen counterexample. In this paper we characterize the optimal control designs for the vector-valued setting assuming that it results in an internal state that can be described by a continuous random variable which has a probability density function. More specifically, we provide a genie-aided outer bound that relies on our previous results for empirical coordination problems. This solution turns out to be not optimal in general, since it consists of a time-sharing strategy between two linear schemes of specific power. It follows that the optimal decision strategy for the original scalar Witsenhausen problem must lead to an internal state that cannot be described by a continuous random variable which has a probability density function.

Locality enables storage systems to recover failed nodes from small subsets of surviving nodes. The setting where nodes are partitioned into subsets, each allowing for local recovery, is well understood. In this work we consider a generalization introduced by Gopalan et al., where, viewing the codewords as arrays, constraints are imposed on the columns and rows in addition to some global constraints. Specifically, we present a generic method of adding such global parity-checks and derive new results on the set of correctable erasure patterns. Finally, we relate the set of correctable erasure patterns in the considered topology to those correctable in tensor-product codes.

A novel technique for digital backpropagation (DBP) in wavelength-division multiplexing systems is introduced and shown, by simulations, to outperform existing DBP techniques for approximately the same complexity.

This work is focused on the system-level performance of a broadcast network. Since all transmitters in a broadcast network transmit the identical signal, received signals from multiple transmitters can be combined to improve system performance. We develop a stochastic geometry based analytical framework to derive the coverage of a typical receiver. We show that there may exist an optimal connectivity radius that maximizes the rate coverage. Our analysis includes the fact that users may have their individual content/advertisement preferences. We assume that there are multiple classes of users with each user class prefers a particular type of content/advertisements and the users will pay the network only when then can see content aligned with their interest. The operator may choose to transmit multiple contents simultaneously to cater more users' interests to increase its revenue. We present revenue models to study the impact of the number of contents on the operator revenue. We consider two scenarios for users' distribution: one where users' interest depends on their geographical location and the one where it doesn't. With the help of numerical results and analysis, we show the impact of various parameters including content granularity, connectivity radius, and rate threshold and present important design insights.