  In network communication, the source often transmits messages at several
different information rates within a session. How to deal with information
transmission and network error correction simultaneously under different rates
is introduced in this paper as a variable-rate network error correction
problem. Apparently, linear network error correction MDS codes are expected to
be used for these different rates. For this purpose, designing a linear network
error correction MDS code based on the existing results for each information
rate is an efficient solution. In order to solve the problem more efficiently,
we present the concept of variable-rate linear network error correction MDS
codes, that is, these linear network error correction MDS codes of different
rates have the same local encoding kernel at each internal node. Further, we
propose an approach to construct such a family of variable-rate network MDS
codes and give an algorithm for efficient implementation. This approach saves
the storage space for each internal node, and resources and time for the
transmission on networks. Moreover, the performance of our proposed algorithm
is analyzed, including the field size, the time complexity, the encoding
complexity at the source node, and the decoding methods. Finally, a random
method is introduced for constructing variable-rate network MDS codes and we
obtain a lower bound on the success probability of this random method, which
shows that this probability will approach to one as the base field size goes to
infinity.
