  We report on a particular example of noise and data representation
interacting to introduce systematic error. Many instruments collect integer
digitized values and appy nonlinear coding, in particular square-root coding,
to compress the data for transfer or downlink; this can introduce surprising
systematic errors when they are decoded for analysis. Square root coding and
subsequent decoding typically introduces a variable, $\pm 1$ count
value-dependent systematic bias in the data after reconstitution. This is
significant when large numbers of measurements (e.g., image pixels) are
averaged together. Using direct modeling of the probabiliity distribution of
particular coded values in the presence of instrument noise, one may apply
Bayes' Theorem to construct a decoding table that reduces this error source to
a very small fraction of a digitizer step; in our example, systematic error
from square root coding is reduced by a factor of 20 from 0.23 count RMS to
0.013 count RMS. The method is suitable both for new experiments such as the
upcoming PUNCH mission, and also for post facto application to existing data
sets -- even if the instrument noise properties are only loosely known.
Further, the method does not depend on the specifics of the coding formula, and
may be applied to other forms of nonlinear coding or representation of data
values.
