# Algebra Seminar

An algebraic framework for end-to-end physical-layer network coding

Abstract: In the framework of physical-layer network-coding (PLNC), multiple terminals attempt to exchange sets of messages through intermediate relay nodes. Recently, Feng, Silva and Kschischang developed an algebraic framework to study PLNC schemes, where messages can be represented as a modules over a finite principal ideal ring.

In this talk, in analogy with random linear network coding, we propose an algebraic framework for modules transmission based on module length. We define a submodule code as a collection of submodules of a given ambient space module, and measure the distance between submodules via a function which we call the submodule distance. Both information loss and errors are captured by the submodule distance.

Using the row-echelon form of a matrix over a principal ideal ring, we reduce the computation of the distance between submodules to the computation of the length of certain ideals in the base ring.

We then present two bounds on the size of a submodule code of given minimum distance and whose codewords have fixed length. For certain classes of rings, we state our bounds explicitly in terms of the ring and code’s invariants. Finally, we construct classes of submodule codes with maximum error-correction capability. In particular, we construct asymptotically optimal codes over certain rings that are relevant from an applied viewpoint.