Ring-theoretic foundation of convolutional network coding

Convolutional network coding deals with the propagation of symbol streams through a network with a linear time-invariant encoder at every node. When the symbol alphabet is a field F, a symbol stream becomes a power series over F. Physical implementation requires the coding/decoding kernels be restricted to finite objects. A proper domain for convolutional network coding consists of rational power series rather than polynomials, because polynomial coding kernels do not necessarily correspond to polynomial decoding kernels when the network includes a cycle. One naturally wonders what algebraic structure makes rational power series a suitable domain for coding/decoding kernels. The proposed answer by this paper is discrete valuation ring (DVR). A general abstract theory of convolutional network coding is formulated over a generic DVR and does not confine convolutional network coding to the combined space-time domain. Abstract generality enhances mathematical elegance, depth of understanding, and adaptability to practical applications. Optimal convolutional network codes at various levels of strength are introduced and constructed for delivering highest possible data rates.