  This is the third paper on the improvements of systematic errors in our weak
lensing analysis using an elliptical weight function, called E-HOLICs. In the
previous papers we have succeeded in avoiding error which depends on
ellipticity of background image. In this paper, we investigate the systematic
error which depends on signal to noise ratio of background image. We find that
the origin of the error is the random count noise which comes from Poisson
noise of sky counts. Random count noise makes additional moments and centroid
shift error, and those 1st orders are canceled in averaging, but 2nd orders are
not canceled. We derived the equations which corrects these effects in
measuring moments and ellipticity of the image and test their validity using
simulation image. We find that the systematic error becomes less than 1% in the
measured ellipticity for objects with $S/N>3$.
