Welch's Bound and Sequence Sets for Code-Division Multiple-Access Systems

Welch's bound for a set of M complex equi-energy sequences is considered as a lower bound on the sum of the squares of the magnitudes of the inner products between all pairs of these sequences. It is shown that, when the sequences are binary (±-1 valued) sequences assigned to the M users in a synchronous code-division multiple-access (S-CDMA) system, precisely such a sum determines the sum of the variances of the interuser interference seen by the individual users. It is further shown that Welch's bound, in the general case, holds with equality if and only if the array having the M sequences as rows has orthogonal and equi-energy columns. For the case of binary (±-1 valued) sequences that meet Welch's bound with equality, it is shown that the sequences are uniformly good in the sense that, when used in a SCDMA system, the variance of the interuser interference is the same for all users. It is proved that a sequence set corresponding to a binary linear code achieves Welch's bound with equality if and only if the dual code contains no codewords of Hamming weight two. Transformations and combination of sequences sets that preserve equality in Welch's bound are given and used to illustrate the design and analysis of sequence sets for non-synchronous CDMA systems.