  In the practical network communications, many internal nodes in the network
are required to not only transmit messages but decode source messages. For
different applications, four important classes of linear network codes in
network coding theory, i.e., linear multicast, linear broadcast, linear
dispersion, and generic network codes, have been studied extensively. More
generally, when channels of communication networks are noisy, information
transmission and error correction have to be under consideration
simultaneously, and thus these four classes of linear network codes are
generalized to linear network error correction (LNEC) coding, and we say them
LNEC multicast, broadcast, dispersion, and generic codes, respectively.
Furthermore, in order to characterize their efficiency of information
transmission and error correction, we propose the (weakly, strongly) extended
Singleton bounds for them, and define the corresponding optimal codes, i.e.,
LNEC multicast/broadcast/dispersion/generic MDS codes, which satisfy the
corresponding Singleton bounds with equality. The existences of such MDS codes
are discussed in detail by algebraic methods and the constructive algorithms
are also proposed.
