Generalized M -Network: A Case for Fixed Message Dimensions

• Published in: IEEE communications letters 2020-01, Vol.24 (1), p.38-42
• Main Authors: Singh, Vikrant, Zolfaghari, Behrouz, Kumar, Chunduri Venkata Dheeraj, Rai, Brijesh Kumar, Bibak, Khodakhast, Srivastava, Gautam, Roy, Swapnoneel, Koshiba, Takeshi
• Format: Article
• Language: English
• Subjects: Computer science; fixed message dimension; Generalized <italic xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">M -Networks; Linear codes; linear coding capacity; Network coding; Random processes; Routing
• ISSN: 1089-7798; 1558-2558
• DOI: 10.1109/LCOMM.2019.2950193

In this letter, we first present a class of networks named Generalized M m,r -Network for every integer m ≥ 2 and ∀ r ∈ {0, 1,..., 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 the Generalized M-Network presented in the work of Das and Rai and the Dim-m Network introduced in the work of Connelly and Zeger which are generalizations to the M-Network can be considered as special cases of Generalized M 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 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 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 m,r -Network, our studies pose some open problems which make the Generalized M m,r -Network an attractive topic for further research.