The efficient encoding and transmission of information for facsimile communication relies on redundancy in the scanned pixels. Halftone images, especially those rendered by high-quality dispersed dot techniques, are "busy" with alternative black and white pixels and shorter run lengths as compared to text information. Because of this, it is desirable to increase the redundancy and decrease the entropy of those images for efficient encoding and transmission. We propose a novel technique whereby both transmitting and receiving fax devices have in memory a halftone screen such as the "blue noise mask" (BNM). The BNM is a halftone screen that produces a visuaily appealing dispersed dot pattern with an unstructured, isotropic pattern. When both the transmitting and receiving fax devices have the same halftone screen in ROM, the problem of halftone image encoding can be reduced to that of transmitting the mean gray value of blocks, or subimages, followed by a sparse halftone error image with increased redundancy and run-lengths compared to the original halftone. Examples show that by using the proposed technique, image entropy can be reduced to 0.2 bits/pixel, and typical run-lengths can increase by a factor of 5. The increase in image quality, combined with increased transmission speed, could add considerably to the utilization and acceptance of halftone fax images.