A linear network (LN) code can be described by either global encoding kernels (GEKs) or local encoding kernels (LEKs). In the literature, the multicast property of an LN code is described using GEKs, so the design algorithms for multicast LN codes employ GEKs to check this property. In this paper, a criterion is developed so that LEKs rather than GEKs can be used to check the multicast maximum distance separable (MDS) property of linear network error correction (LNEC) codes on erroneous networks. Further, it is used to develop a design algorithm for multicast MDS LNEC codes on erroneous cyclic networks. This algorithm is more efficient than the algorithm that uses GEKs when the number of edges is high and the interconnection of these edges is low.

Recently, the refined singleton bound over acyclic networks was extended to convolutional error-correcting network codes over cyclic networks using extended coding vectors. In this paper, it is shown that constructing an MDS code is equivalent to constructing a multicast code. This is used to develop an algorithm for constructing MDS field-based codes over acyclic networks when the sinks know the topology of the network and the network coding employed at all nodes. A lower bound is given on the size of the field required for the algorithm to be successful. Then this algorithm is extended to construct MDS convolutional error-correcting codes over cyclic networks. The complexity of the proposed algorithm is evaluated.

In this study, the authors study ring‐based linear network coding over erroneous cyclic networks over commutative rings such as a principal ideal domain or discrete valuation ring. In the first part, they study coherent field‐based linear error‐correcting network codes (LENCs) over cyclic networks. By changing alphabet symbols from fields to commutative rings, they extend Zhang's formulation for LENCs restricted on acyclic networks to cyclic networks, and generalise fundamental results and concepts such as the minimum rank distance and the refined Singleton bound, and show that this bound is tight. In the second part, they generalise some main results such as the free distance and the generalised Singleton bound from convolutional codes to ring‐based LENCs over cyclic networks. In the third part, they propose an algebraic method for calculating sink bit error probability of ring‐based linear network codes over all erroneous networks by using the authors’ formulations.