  Recently, network error correction coding (NEC) has been studied extensively.
Several bounds in classical coding theory have been extended to network error
correction coding, especially the Singleton bound. In this paper, following the
research line using the extended global encoding kernels proposed in
\cite{zhang-correction}, the refined Singleton bound of NEC can be proved more
explicitly. Moreover, we give a constructive proof of the attainability of this
bound and indicate that the required field size for the existence of network
maximum distance separable (MDS) codes can become smaller further. By this
proof, an algorithm is proposed to construct general linear network error
correction codes including the linear network error correction MDS codes.
Finally, we study the error correction capability of random linear network
error correction coding. Motivated partly by the performance analysis of random
linear network coding \cite{Ho-etc-random}, we evaluate the different failure
probabilities defined in this paper in order to analyze the performance of
random linear network error correction coding. Several upper bounds on these
probabilities are obtained and they show that these probabilities will approach
to zero as the size of the base field goes to infinity. Using these upper
bounds, we slightly improve on the probability mass function of the minimum
distance of random linear network error correction codes in
\cite{zhang-random}, as well as the upper bound on the field size required for
the existence of linear network error correction codes with degradation at most
$d$.
