Generalized $M_{m,r}$ -Network: A Case for Fixed Message Dimensions
V.K. SINGH, Behrouz Zolfaghari, Chunduri Venkata Dheeraj Kumar, Brijesh Kumar Rai, Khodakhast Bibak, Gautam Srivastava, Swapnoneel Roy, Takeshi Koshiba · IEEE Communications Letters · 2019
In this letter, we first present a class of networks namedGeneralized${{M}}_{{ \textit {m, r}}}$-Networkfor every integer${m} \geq 2$and$\forall {r} \in \{0, 1,\ldots, {m}-1\}$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${m}$. We show that theGeneralized${M}$-Networkpresented in the work of Das and Rai and theDim-${m}$Networkintroduced in the work of Connelly and Zeger which are generalizations to the${M}$-Network can be considered as special cases of Generalized${M}_{\textit {m, r}}$-Network for${r}=1$and${r}={m}-1$respectively. Then we focus on a problem induced by depending on integer multiples of${m}$as message dimensions to achieve the linear coding capacity in the class of Generalized${M}_{\textit {m, r}}$(proven to be equal to 1). We note that for large values of${m}$, 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${M}_{\textit {m, r}}$-Network highlights the importance of the examined problem. In addition to complete/partial solutions obtained for different networks of the class Generalized${M}_{\textit {m, r}}$-Network, our studies pose some open problems which make the Generalized${M}_{\textit {m, r}}$-Network an attractive topic for further research.