My comment about digital cameras with high pixel counts was a little misleading.
Without becoming too technical; digital noise is more closely related to the ASA speed that you use to take pictures. For low light photography you would normally use a higher ASA rating than you would in good light.
To achieve that higher speed, the camera software amplifies the electrical signal from the photosites on the sensor. Along with the useful signal that you want, the background signal is also amplified. It is similar to increasing the volume on a cheap stereo - at high levels you start to hear the hiss.
Digital noise in photos is also related to longer exposure times and is a product of the heat that is generated on the sensor. This problem can also be caused by placing a large bright viewing screen just behind the sensor. The heat generated by the screen can increase the noise in the photos unless care is taken to isolate the two components. Leaving your camera in the glove compartment of a car on a hot day causes a similar problem.
The smaller sensors in compact cameras are more prone to noise. That is because the smaller photosites on a small sensor produce a lot less current than the correspondingly larger photosites on a big sensor. They therefore require a greater degree of amplification to produce an image.
Now to the issue of cramming more pixels onto a small sensor. Cramming more photosites onto a small sensor e.g on a high megapixel compact camera, not only increases the amplification required but also increases the contamination between neighbouring photosites which further adds to the digital noise.
So my explanation was only part of the picture - I apologise for that.
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