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Kailh's most common low-profile switch is called "Kailh Choc".
They are incompatible with Cherry MX because they have the pins at the bottom in a different arrangement. In other words, they will not physically fit PCB made only for Cherry MX-style switches.
There are a bunch of PCBs out there for Kailh Choch, and even some that can actually fit both Kailh Choc or Cherry MX. They tend to be sold by small stores that cater exclusively to the enthusiast market, and in group orders here on Geekhack.
Some of the PCB designs are "open source"/"open hardware", and the schemas can be found on Github and other free repository sites.
However ... most of these PCBs are for layouts that are minimalist, columnar and/or ergonomic though. One contributing reason for this is that the keycaps that Kailh makes for them have unusual non-standard sizes for some inexplicable reason. (both in the size of a 1×1 unit key and the number of units wide some are)
I have heard of a new keycap profile coming, but they have the same restrictions as Kailh's.
Only mainstream ready-made keyboards with Kailh Choc switches come with keys in standard sizes and layouts.
There are dozens of stores out there but most that do sell only a single kit. From the top of my head, I can mention keeb.io. keycapsss.com. splitkb.com.
The Japanese store yushakobo.jp has a lot — minimalist low-profile PCBs seem to be very popular in the keyboard scene over there.
In a couple years time, I think we will have moved away from Kailh Choc though, and towards Cherry MX Low Profile and clones. Neither of these are pin-compatible with Cherry MX either though, despite the name (which Cherry had chosen for marketing more than anything else...)
These fit most keycaps for Cherry MX, if the keyboard does not have a plate above the PCB. Kailh's "Choc V2" has the same shape and size as Cherry MXLP but a different pinout, and TTC is making a clone that is pin-compatible. The clones are not generally available and it's a bit of a chicken-and-the-egg situation for stores if there are so few PCBs available.
What kind of keyboard are you looking at building?
