22 Bench Rest (BR) Remington, or could be 6mm BR Remington or 7mm BR Remington.
What length are the brass? Remington also made 308 cases for a while to be formed into BR Brass. It was normal 308 brass with a small rifle primer and IRRC was just marked BR Remington. Bench rest shooters are typically very picky and methodical about the brass. From the 308 small primer cases they could make it into the BR cartridge of choice and turn the necks for their rifles. The 7mmBR was also used a lot in the old XP-100 pistol.