Hello simon,
I can't remember running across anything specific on growing hydroponic root crops in general. However I might be able to find something on a specific crop/s. I don't have time right now so I will try to look later. But for now as far as what to grow it in. My understanding is that root crops do best in
growing medium like sand, or sand and vermiculite/perlite mix, and using a
drip system. Though I have seen that flood and drain systems can work also, but be careful with the water level not getting too high, or you could wind up with having a liquefaction problem. That is if the water level is to to high it can make the sand/mixture unstable for the plants and they can fall over. Personally I would go with a drip system for root crops (I haven't grown any yet). Much easier in design, control, and allowing the water to filter down without the sand going with it.
P.S. Just make sure you wash and sanitize the sand first. Also I have seen that coco coir has been used as well, but I think it's to light and can be a problem with odd shaped veggies, though still editable.