In Analog-to-digital (A/D) conversion, signal decimation has been proven to greatly improve the efficiency of data storage while maintaining high accuracy. When one couples signal decimation with the $ΣΔ$ quantization scheme, the reconstruction error decays exponentially with respect to the bit-rate. We build on our previous result, which extends signal decimation to finite frames, albeit only up to the second order. In this study, we introduce a new scheme called adapted decimation, which yields polynomial reconstruction error decay rate of arbitrary order with respect to the oversampling rate, and exponential with respect to the bit-rate.