Programmable gain instrumentation amplifier with R-2R ladder and analog multiplexer

I want to build a programmable gain instrumentation amplifier with power of two possible gains (1x, 2x, 4x, 8x...). Is there a way to use a R-2R resistor ladder and an analog multiplexer in order to tune the $R_{gain}$ ?

On the web I found this way to make programmable gain amplifiers using a R-2R ladder and an analog multiplexer but I don't want to attenuate my input signal as it is already small, I'd lose it in the noise in this way.