Please provide me the source image and the result you get. You can use http://bugs.libgd.org as well as you can add attachments to your report.

no need for src images :-)

In fact yes, I need the source image (what you have in $imageData). Sorry for the double posts.

We need a better reproducible example here, including the source image. When the source image has a perfect alpha channel, I see no noise. Try this:
$imageSrc = imagecreatetruecolor(500, 500);
imagealphablending($imageSrc, false);
imagesavealpha($imageSrc, true);
$color = imagecolorallocatealpha($imageSrc, 0, 0, 0, 127);
imagefill($imageSrc, 0, 0, $color);
$imageDst = imagecreatetruecolor(50, 50);
imagealphablending($imageDst, false);
imagesavealpha($imageDst, true);
imagecopyresampled($imageDst,$imageSrc, 0, 0, 0, 0, 50, 50, 500, 500);
header('Content-type: image/png');
imagepng($imageDst);

Please try the following with different image sizes and you will see that the alpha channel has noise.
(I have two original before and after images but can not find the link to upload them as attachments)
$imageSrc = imagecreatetruecolor(90, 90);
imagealphablending($imageSrc, false);
imagesavealpha($imageSrc, true);
$color = imagecolorallocatealpha($imageSrc, 0, 0, 0, 127);
imagefill($imageSrc, 0, 0, $color);
$imageDst = imagecreatetruecolor(56, 56);
imagealphablending($imageDst, false);
imagesavealpha($imageDst, true);
imagecopyresampled($imageDst,$imageSrc, 0, 0, 0, 0, 56, 56, 90, 90);
header('Content-type: image/png');
imagepng($imageDst);

Here are my test source image (png 90x90): http://tinyurl.com/65tdtm
Resized to 56x56 with noise in alpha channel: http://tinyurl.com/6yold7

The 56x56 image looks perfectly fine to me. I also checked the alpha values for the translucent areas and they look fine too (full translucent when they are not near the star anti aliased pixels, as expected).
To see how is the alpha channel in the result image, please look at this zoomed version of the alpha channel only:
http://pierre.libgd.org/bugs/45030_alpha_only.png
100% black means opaque and the gray squares are only to show a background (what would be behind the image). As you can see, there is no noise in the transparent areas.
Are you sure that there is not a bug in the Motorola display system? Maybe it does not support semi transparent pixels.

You can see the noise on both my 56x56 image and image generated from Rasmus's reproducing code sample with resizing from 90x90 to 50x50 (I am not talking about the anti-aliasing areas :)
GIMP: Tools > Selection Tools > By Color select, click a few spots in the empty region of the image and you will see the noise being selected.
Paint.net: use the color select tool and click a few times on the empty region of the image, some pixels will display the transparency as 2.
(thank you so much for your quick response!)

Right, there is a alpha values between 0 and 2. This is a little rounding issue in the interpolation function.
As it is not relevant when you use the full range of the 8bit (7bit in gd 2.0.x internals), it could cause some troubles when you introduce more errors while using only 4bits. 
The worst case will end with an error twice bigger than the original one. That's certainly why you see the little black pixels.
Here is an attempt to minimize the error in the edge of the alpha values:
http://pierre.libgd.org/patches/bug45030.txt
It should fix non obvious errors like the one you had.

We have verified the fix resolved our problem. Thank you very much! Do you have any estimate on when the fix will make into the stable branch?

Pierre, the fix worked, commit?

Applied.