Bounds on the Number of Signals with Restricted Cross Correlation

The author seeks the maximum number of users possible for multiuser, spread-spectrum radio systems, as the signal set is varied, under the following ideal conditions: 1) no fading; 2) synchronous system; 3) orthogonal signal set for each user; 4) matched-filter individual receivers; 5) no additive noise. Welch (1974) has given an upper bound on the number of vectors (or equivalently signals) w possible in a d-dimensional space, given the maximum cross-correlation magnitude C_max between any vector pair. However, the error rate for the systems under study may depend on a suitably defined root-mean-square cross correlation C_rms, rather than on the maximum cross correlation C_max of a vector set. Welch´s results are extended to d-dimensional vector sets of size w divided into alphabets of size A, the vectors of each alphabet being strictly orthogonal, and the maximum size of such a vector set is given as function of C_rms. It is demonstrated that as Welch´s limit is approached, all cross correlations must approach C_max, and given precise limits on the way in which this must occur. Similarly, close to the present bounds (in terms of C_rms) all rms cross correlations must be close to C_rms. Particular examples are given of signal sets close to the present bounds.