Ring-based linear network coding on erroneous cyclic networks

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.