Many components of modern infrastructure such as transportation systems, power systems, climate and environment monitoring systems, education systems and even government are being increasingly interconnected through information networks. Information is playing an ever bigger role in our lives than just a decade ago.There is a need to gather, store, process and communicate information across  several distributed devices. These devices continually and simultaneously perform one or more of these information processing tasks. In doing so, they share resources, co-ordinate to achieve their objectives,  and even at times overcome individual constraints through the pooling of networked resources. Central to the functioning of current day information networks are strategies that facilitate these network information processing objectives. In this project, we have addressed the overarching challenge of designing efficient information processing strategies from a fundamental network information theory viewpoint. 

Characterizations of fundamental performance limits in communication over networks  require  devising optimal transmitters and receivers at distributed network terminals. Mathematically, the operations of transmitters and receivers are characterized using encoding functions and decoding functions, respectively, also referred to as codes.The image of an encoding function in channel coding and a decoding function in source coding is referred to as a codebook.  Hence, one of the main objectives in network information theory is to design optimal codes. The conventional technique of deriving the performance limits for any communication problem in information theory is  via random coding involving so-called  Independent Identically Distributed (IID) random codebooks. 

In this project we have constructured structured code ensembles.  Firstly, we have proposed two new ensembles of coset codes (shifted linear codes): (i) partitioned coset codes (PCC), and (ii) unionized coset codes (UCC). These codes possess both the empirical properties present in unstructured codes as well as algebraic closure properties. A PCC is a coset code that is randomly partitioned into bins. This means that each codeword in the shifted linear code is assigned a bin number chosen randomly and uniformly among a predetermined number of bins. Absent such partitioning, linear codes can only achieve a very limited set of empirical distributions (e.g. uniform distributions). The partitioning ensures each bin possesses codewords with a specified empirical distribution and therebyachieves rates corresponding to non-uniform distributions. The overall code being a coset code possess (algebraic) closure properties. 

On the other hand, a UCC is a collection of arbitrary cosets comprising a code. Similar to the binning operation in PCCs, the unionizing operation in UCCs allows them to achieve non-uniform empirical distributions. In many multiterminal communication systems, we see that both covering codes and packing codes are necessary in order to achieve the optimal performance. Moreover, they are used such that either a packing code is partitioned into covering codes or the other way around.  In other words, we have nested codes with a denser code (packing/covering) containing a sparser code (covering/packing). As we have seen,  one can endow asymptotically optimal  codes with the algebraic structure associated with a finite field with no cost. In other words, finite field structure comes for free.

Secondly, in this project, we have built upon the technique of typicality set encoding and decoding  to generalize coding techniques to arbitrary sources and channels. It can be noted that the theory developed in this framework can be further generalized to continuous-valued sources and channels, and in particular Gaussianmulti-terminal channels and sources. Lattices have been studied extensively for source coding and channel coding in the linear quadratic Gaussian setting both in the point-to-point and multiterminal settings. In particular, a framework involving lattice codes for communicating over real-valued channels can be obtained by mapping coset codes to lattice codes. These findings indicate that coset codes built over finite fields described here  are only the first step in exploiting algebraic closure properties. Furthermore, it turns out that a framework based on codes built over groups can be employed to derive even larger achievable rate regions. 

The main results obtained in this project were disseminated in premier conferences and journals in the field of network information theory. Through this project, two graduate students received training in teaching and research. The research perfomed through this project were documented in the PhD theses of these students. The principal investigator created a graduate course based on this research titled Network information theory in the University of Michigan. This course is being taught once every two years. This course has educated several talented graduate students in the area of network information theory. Through this course, women and underrepresented minority students were encouraged to participate and excel in research and teaching of information theory.  The project also led to the publication of a research monograph titled ``An algebraic and probabilistic framework for network information theory'' published by NOW publications. 

Last Modified: 01/16/2021
Modified by: Sandeep Sadanandarao