Optimization of signal waveforms

A technique is described whereby it is possible to determine a set of waveforms that minimize the error rate for a digital communication system. The waveforms are generated from a finite set of orthogonal waveforms. Each character waveform has the same energy as any other character waveform. The technique may be applied to any channel where the source and channel statistics are known. It is also applied to the problem of maximizing the minimum distance between a set of points on an -dimensional sphere. The solution of this problem can be used to minimize the error rate for a system disturbed by white Gaussian noise of very low average noise power per unit of bandwidth. In particular a possible solution for the spacing of ten points on a four-dimensional sphere is shown. It represents a considerable improvement over the best previous result.