I am a desert located primarily in one country, bordered by a large river to my east, mountains to my west, and known for my unique 'fairy circles'. My sand is unusually red due to high iron oxide content. What desert am I?

1. I am named after a local word meaning 'vast place'.
2. I am largely uninhabited except for scattered nomadic groups.
3. Diamonds have been found beneath my sands.
4. My climate is hyper-arid with very little rainfall.