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We generalize the total square correlation and the total weighted square correlation (TWSC) for a given signature set used in synchronous code division multiple access (S-CDMA) systems. We define the extended TWSC measure for multipath channels in the presence and the absence of the colored noise. The main results of this paper are the following: 1, the necessary and sufficient conditions of the received Gram matrix to maximize the sum capacity in the presence of multipath; 2, the conditions on the channel such that the received multipath sequences in the presence of the colored noise are Welch bound equality (WBE) sequences; 3, a decentralized method for obtaining generalized WBE sequences that minimize the TWSC in the presence of multipath and colored noise. Using this method, the optimal sequences, which achieve sum capacity for overloaded S-CDMA systems, in the presence of multipath are obtained. Numerical examples that illustrate the mathematical formalism are also included.