A geometric approach to dynamic network coding

Abstract

Subspace coding over linear network channels assuming incoherent transmission allows independent design of channel and network codes. Joint design however would be desirable for dynamic network conditions. In this work a geometrical approach (in the Kleinian sense) to dynamic network coding is presented. The approach consists of capturing the communication process with group actions. Specifically, codes are chosen as geometries: homogeneous spaces obtained from group actions carry the information and the dynamic network code is the stabilizer of the action. The approach subsumes other approaches and provides natural adaptive encoding and decoding schemes with linear algebra tractability over different communication ambient spaces. The algebraic object called flag is proposed to encode information while the dynamic network coding is specified by its stabilizer (Borel group) showing the interplay between the flag, the channel impairing the flag and the network code stabilizing the flag. Ergodic capacity achievability is discussed.