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Algebraic Structure in Network Information Theory

Mathematics has always played an important role in the design and optimization of communication systems, particularly following Shannon’s groundbreaking work in 1948 that gave theorems establishing fundamental limits on the rates of reliable communication over point-to-point channels (i.e., those involving a single transmitter-receiver pair). In order to develop fundamental bounds, concepts from probability and statistics and arguments involving averages taken over random code ensembles have been of key importance in most of the communications problems studied to date. Endowing these code ensembles with particular algebraic structure (e.g., the structure of a vector space) was not necessary to establish fundamental limits, and entered the stage only to allow for compact code descriptions and computationally-efficient encoding and decoding algorithms. In networks (i.e., communication systems involving multiple transmitters or receivers or relay nodes) the situation becomes much more complicated, and a general framework that establishes fundamental limits is lacking. The current grand challenge in information theory is to devise communication strategies and architectures that optimally exploit communication networks. An emerging key insight—and the motivation for this workshop—is that in certain network information theory problems, endowing code ensembles with algebraic structure is a key necessity, not only for engineering convenience, but for the derivation of fundamental limits. The goal of the workshop was to bring together experts from information theory, coding theory, and algebra to shed light on this observation, with the goal of beginning to understand the type and extent of algebraic structure needed to extend Shannon’s insights about point-to-point channels to the more general case of networks.