We have invented a new modem technology for transmitting data on conventional telephone lines (twisted pairs) at high speeds. This discovery is timely, as new standards are being developed for this Digital Subscriber Line (DSL) technology at this very moment. The potential market for the new modem technology is massive, as the telephone service providers wish to offer Internet access to the masses using the current phone lines into the home. Key to the deployment of any new service is the distribution of power over frequency, for new services must be designed to be robust to interference that might be caused by other services that are carried by neighboring telephone lines. As well, new services cannot interfere with existing services. We have made two discoveries. The first is an optimization technique that provides the best possible distribution of power (over frequency) for any new DSL service given the interference from other known services that are carried by neighboring telephone lines in the same cable. The second is a power istribution scheme that minimizes the interference caused by the new DSL service into neighboring lines. This new modem technology can be applied to many channels besides the telephone channel (for example, coaxial cables, power lines, wireless channels, and telemetry cables used in geophysical well-logging tools).