An Upper Bound on the Sizes of Multiset-Union-Free Families

Let $\mathcal{F}_1$ and $\mathcal{F}_2$ be two families of subsets of an $n$-element set. We say that $\mathcal{F}_1$ and $\mathcal{F}_2$ are multiset-union-free if for any $A,B\in \mathcal{F}_1$ and $C,D\in \mathcal{F}_2$ the multisets $A\uplus C$ and $B\uplus D$ are different, unless both $A = B$ and $C= D$. We derive a new upper bound on the maximal sizes of multiset-union-free pairs, improving a result of Urbanke and Li.Comment: A shorter ISIT conference version titled "VC-Dimension Based Outer Bound on the Zero-Error Capacity of the Binary Adder Channel" is availabl

Applications of Coding Theory to Massive Multiple Access and Big Data Problems

The broad theme of this dissertation is design of schemes that admit iterative algorithms with low computational complexity to some new problems arising in massive multiple access and big data. Although bipartite Tanner graphs and low-complexity iterative algorithms such as peeling and message passing decoders are very popular in the channel coding literature they are not as widely used in the respective areas of study and this dissertation serves as an important step in that direction to bridge that gap. The contributions of this dissertation can be categorized into the following three parts. In the first part of this dissertation, a timely and interesting multiple access problem for a massive number of uncoordinated devices is considered wherein the base station is interested only in recovering the list of messages without regard to the identity of the respective sources. A coding scheme with polynomial encoding and decoding complexities is proposed for this problem, the two main features of which are (i) design of a close-to-optimal coding scheme for the T-user Gaussian multiple access channel and (ii) successive interference cancellation decoder. The proposed coding scheme not only improves on the performance of the previously best known coding scheme by ≈ 13 dB but is only ≈ 6 dB away from the random Gaussian coding information rate. In the second part construction-D lattices are constructed where the underlying linear codes are nested binary spatially-coupled low-density parity-check codes (SCLDPC) codes with uniform left and right degrees. It is shown that the proposed lattices achieve the Poltyrev limit under multistage belief propagation decoding. Leveraging this result lattice codes constructed from these lattices are applied to the three user symmetric interference channel. For channel gains within 0.39 dB from the very strong interference regime, the proposed lattice coding scheme with the iterative belief propagation decoder, for target error rates of ≈ 10^-5, is only 2:6 dB away the Shannon limit. The third part focuses on support recovery in compressed sensing and the nonadaptive group testing (GT) problems. Prior to this work, sensing schemes based on left-regular sparse bipartite graphs and iterative recovery algorithms based on peeling decoder were proposed for the above problems. These schemes require O(K logN) and Ω(K logK logN) measurements respectively to recover the sparse signal with high probability (w.h.p), where N, K denote the dimension and sparsity of the signal respectively (K (double backward arrow) N). Also the number of measurements required to recover at least (1 - €) fraction of defective items w.h.p (approximate GT) is shown to be cv€_K logN/K. In this dissertation, instead of the left-regular bipartite graphs, left-and- right regular bipartite graph based sensing schemes are analyzed. It is shown that this design strategy enables to achieve superior and sharper results. For the support recovery problem, the number of measurements is reduced to the optimal lower bound of Ω (K log N/K). Similarly for the approximate GT, proposed scheme only requires c€_K log N/ K measurements. For the probabilistic GT, proposed scheme requires (K logK log vN/ K) measurements which is only log K factor away from the best known lower bound of Ω (K log N/ K). Apart from the asymptotic regime, the proposed schemes also demonstrate significant improvement in the required number of measurements for finite values of K, N

Applications of Coding Theory to Massive Multiple Access and Big Data Problems

The broad theme of this dissertation is design of schemes that admit iterative algorithms with low computational complexity to some new problems arising in massive multiple access and big data. Although bipartite Tanner graphs and low-complexity iterative algorithms such as peeling and message passing decoders are very popular in the channel coding literature they are not as widely used in the respective areas of study and this dissertation serves as an important step in that direction to bridge that gap. The contributions of this dissertation can be categorized into the following three parts. In the first part of this dissertation, a timely and interesting multiple access problem for a massive number of uncoordinated devices is considered wherein the base station is interested only in recovering the list of messages without regard to the identity of the respective sources. A coding scheme with polynomial encoding and decoding complexities is proposed for this problem, the two main features of which are (i) design of a close-to-optimal coding scheme for the T-user Gaussian multiple access channel and (ii) successive interference cancellation decoder. The proposed coding scheme not only improves on the performance of the previously best known coding scheme by ≈ 13 dB but is only ≈ 6 dB away from the random Gaussian coding information rate. In the second part construction-D lattices are constructed where the underlying linear codes are nested binary spatially-coupled low-density parity-check codes (SCLDPC) codes with uniform left and right degrees. It is shown that the proposed lattices achieve the Poltyrev limit under multistage belief propagation decoding. Leveraging this result lattice codes constructed from these lattices are applied to the three user symmetric interference channel. For channel gains within 0.39 dB from the very strong interference regime, the proposed lattice coding scheme with the iterative belief propagation decoder, for target error rates of ≈ 10^-5, is only 2:6 dB away the Shannon limit. The third part focuses on support recovery in compressed sensing and the nonadaptive group testing (GT) problems. Prior to this work, sensing schemes based on left-regular sparse bipartite graphs and iterative recovery algorithms based on peeling decoder were proposed for the above problems. These schemes require O(K logN) and Ω(K logK logN) measurements respectively to recover the sparse signal with high probability (w.h.p), where N, K denote the dimension and sparsity of the signal respectively (K (double backward arrow) N). Also the number of measurements required to recover at least (1 - €) fraction of defective items w.h.p (approximate GT) is shown to be cv€_K logN/K. In this dissertation, instead of the left-regular bipartite graphs, left-and- right regular bipartite graph based sensing schemes are analyzed. It is shown that this design strategy enables to achieve superior and sharper results. For the support recovery problem, the number of measurements is reduced to the optimal lower bound of Ω (K log N/K). Similarly for the approximate GT, proposed scheme only requires c€_K log N/ K measurements. For the probabilistic GT, proposed scheme requires (K logK log vN/ K) measurements which is only log K factor away from the best known lower bound of Ω (K log N/ K). Apart from the asymptotic regime, the proposed schemes also demonstrate significant improvement in the required number of measurements for finite values of K, N

Capacity Theorems for Quantum Multiple Access Channels: Classical-Quantum and Quantum-Quantum Capacity Regions

We consider quantum channels with two senders and one receiver. For an arbitrary such channel, we give multi-letter characterizations of two different two-dimensional capacity regions. The first region is comprised of the rates at which it is possible for one sender to send classical information, while the other sends quantum information. The second region consists of the rates at which each sender can send quantum information. For each region, we give an example of a channel for which the corresponding region has a single-letter description. One of our examples relies on a new result proved here, perhaps of independent interest, stating that the coherent information over any degradable channel is concave in the input density operator. We conclude with connections to other work and a discussion on generalizations where each user simultaneously sends classical and quantum information.Comment: 38 pages, 1 figure. Fixed typos, added new example. Submitted to IEEE Tranactions on Information Theor

Function Computation over Networks:Efficient Information Processing for Cache and Sensor Applications

This thesis looks at efficient information processing for two network applications: content delivery with caching and collecting summary statistics in wireless sensor networks. Both applications are studied under the same paradigm: function computation over networks, where distributed source nodes cooperatively communicate some functions of individual observations to one or multiple destinations. One approach that always works is to convey all observations and then let the destinations compute the desired functions by themselves. However, if the available communication resources are limited, then revealing less unwanted information becomes critical. Centered on this goal, this thesis develops new coding schemes using information-theoretic tools. The first part of this thesis focuses on content delivery with caching. Caching is a technique that facilitates reallocation of communication resources in order to avoid network congestion during peak-traffic times. An information-theoretic model, termed sequential coding for computing, is proposed to analyze the potential gains offered by the caching technique. For the single-user case, the proposed framework succeeds in verifying the optimality of some simple caching strategies and in providing guidance towards optimal caching strategies. For the two-user case, five representative subproblems are considered, which draw connections with classic source coding problems including the Gray-Wyner system, successive refinement, and the Kaspi/Heegard-Berger problem. Afterwards, the problem of distributed computing with successive refinement is considered. It is shown that if full data recovery is required in the second stage of successive refinement, then any information acquired in the first stage will be useful later in the second stage. The second part of this thesis looks at the collection of summary statistics in wireless sensor networks. Summary statistics include arithmetic mean, median, standard deviation, etc, and they belong to the class of symmetric functions. This thesis develops arithmetic computation coding in order to efficiently perform in-network computation for weighted arithmetic sums and symmetric functions. The developed arithmetic computation coding increases the achievable computation rate from $\Theta((\log L)/L)$ to $\Theta(1/\log L)$, where $L$ is the number of sensors. Finally, this thesis demonstrates that interaction among sensors is beneficial for computation of type-threshold functions, e.g., the maximum and the indicator function, and that a non-vanishing computation rate is achievable

A study of major coding techniques for digital communication Final report

Coding techniques for digital communication channel

Concatenation of convolutional and block codes Final report

Comparison of concatenated and sequential decoding systems and convolutional code structural propertie