  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.
