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Double Bed + Tuck = Fabulous!
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What is Tuck Rib?

Tuck Rib always meant Tuck Rib | Fisherman Rib | English Rib to me.

But what about tuck on the main bed, (using automatic patterning) combined with Full Needle Rib

I've seen this technique referred to as Punch Tuck Rib. I believe this was a Singer/Studio/Silver term. No matter what you call it you probably can knit this technique on your automatic patterning machine (punchcard or electronics) with a ribber.

There are 2 prerequisites to successfully knitting Punch Tuck Rib:

Full Needle Rib
You must understand the needle setup for full needle rib
Tuck with automatic patterning (punchcard or electronic)
Practice Tuck on the main bed before putting all needles into work

Standard gauge machine.

Standard gauge machine. Much thinner yarn, fabric is lightweight and open