### Abstract

In this letter, we first present a class of networks named Generalized {{M}}_{{ \textit {m, r}}} -Network for 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 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}_{\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.

Original language | English |
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Article number | 8886606 |

Pages (from-to) | 38-42 |

Number of pages | 5 |

Journal | IEEE Communications Letters |

Volume | 24 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2020 Jan |

### Keywords

- fixed message dimension
- Generalized M-Networks
- linear coding capacity
- Network coding

### ASJC Scopus subject areas

- Modelling and Simulation
- Computer Science Applications
- Electrical and Electronic Engineering

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## Cite this

*IEEE Communications Letters*,

*24*(1), 38-42. [8886606]. https://doi.org/10.1109/LCOMM.2019.2950193