# On the Degrees-of-Freedom of the MIMO Three-Way Channel with Intermittent Connectivity

## TL;DR

This paper investigates the degrees-of-freedom in a MIMO three-way channel with intermittent connectivity, showing that adaptation is essential for achieving the full DoF region, despite nonadaptive schemes reaching the sum-DoF.

## Contribution

It introduces a nonadaptive scheme for the intermittent channel, derives its DoF region, and demonstrates the necessity of adaptation for full DoF region achievement.

## Key findings

- Nonadaptive scheme achieves sum-DoF of the intermittent channel.
- Adaptation is necessary for the full DoF region.
- Counterexample shows adaptive schemes can outperform nonadaptive bounds.

## Abstract

The degrees-of-freedom (DoF) of the multi-antenna three-way channel (3WC) with an intermittent node is studied. Special attention is given to the impact of adaptation. A nonadaptive transmission scheme based on interference alignment, zero-forcing, and erasure-channel treatment is proposed, and its corresponding DoF region is derived. Then, it is shown that this scheme achieves the sum-DoF of the intermittent channel, in addition to the DoF region of the nonintermittent one. Thus, adaptation is not necessary from those perspectives. To the contrary, it is shown that adaptation is necessary for achieving the DoF region of the intermittent case. This is shown by deriving an outer bound for the intermittent channel with nonadaptive encoding, and giving a counterexample of an adaptive scheme which achieves DoF tuples outside this bound. This highlights the importance of cooperation in this intermittent network.