Convolutional Network Coding Based on Matrix Power Series Representation

TL;DR

This paper introduces a matrix power series approach to convolutional network coding, providing theoretical foundations for design and decoding, with implications for practical and wireless network applications.

Contribution

It formulates convolutional network coding using matrix power series, simplifying design with nilpotent matrices and proposing an efficient decoding scheme based on finite terms.

Findings

Unique determination of GEKs by LEK with nilpotent $K_0$

Decodability checked using first $L+1$ terms of $F(z)$

Decoding scheme reduces complexity and suits wireless networks

Abstract

In this paper, convolutional network coding is formulated by means of matrix power series representation of the local encoding kernel (LEK) matrices and global encoding kernel (GEK) matrices to establish its theoretical fundamentals for practical implementations. From the encoding perspective, the GEKs of a convolutional network code (CNC) are shown to be uniquely determined by its LEK matrix $K(z)$ if $K_0$, the constant coefficient matrix of $K(z)$, is nilpotent. This will simplify the CNC design because a nilpotent $K_0$ suffices to guarantee a unique set of GEKs. Besides, the relation between coding topology and $K(z)$ is also discussed. From the decoding perspective, the main theme is to justify that the first $L+1$ terms of the GEK matrix $F(z)$ at a sink $r$ suffice to check whether the code is decodable at $r$ with delay $L$ and to start decoding if so. The concomitant decoding scheme avoids dealing with $F(z)$, which may contain infinite terms, as a whole and hence reduces the complexity of decodability check. It potentially makes CNCs applicable to wireless networks.