Some Connections Between Classical Coding and Network Coding Over Erroneous Cyclic Networks

Recently, a framework was given for linear error-correcting network codes (LENCs) over cyclic networks on commutative rings. When the alphabet is considered as a rational power series ring, an LENC is referred to as a convolutional error-correcting network code (CENC). Recently, a metric was introduced for these codes based on the minimum rank distance. In this paper, a new metric is introduced for ring-based LENCs over cyclic networks based on the Hamming distance, which is referred to as the network Hamming distance. Then, some connections between maximum distance separable (MDS) LENCs and classical MDS codes are obtained. Finally, the network Hamming free distance is given for CENCs, which plays the role of the free distance for convolutional codes.