Generalized [Formula Omitted]-Network: A Case for Fixed Message Dimensions

In this letter, we first present a class of networks named Generalized [Formula Omitted] -Network for every integer [Formula Omitted] and [Formula Omitted] and we show that every network of this class admits a vector linear solution if and only if the message dimension is an integer multiple of [Formula Omitted]. We show that the Generalized [Formula Omitted] -Network presented in the work of Das and Rai and the Dim- [Formula Omitted] Network introduced in the work of Connelly and Zeger which are generalizations to the [Formula Omitted]-Network can be considered as special cases of Generalized [Formula Omitted]-Network for [Formula Omitted] and [Formula Omitted] respectively. Then we focus on a problem induced by depending on integer multiples of [Formula Omitted] as message dimensions to achieve the linear coding capacity in the class of Generalized [Formula Omitted] (proven to be equal to 1). We note that for large values of [Formula Omitted], packet sizes will grow beyond feasible thresholds in real-world networks. This motivates us to examine the capacity of the network in the case of fixed message dimensions. A study on the contrast among the impacts of fixed message dimensions in different networks of class [Formula Omitted]-Network highlights the importance of the examined problem. In addition to complete/partial solutions obtained for different networks of the class Generalized [Formula Omitted]-Network, our studies pose some open problems which make the Generalized [Formula Omitted]-Network an attractive topic for further research.