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It has been shown that signature (multi) sets that meet Welch's lower bound on the total squared correlation (TSC), maximize the sum capacity of the synchronous code division multiple access (CDMA) channel with equal average input energy constraints. However, a signature set that is TSC-optimum (i.e. achieves the lower bound on TSC), doesn't necessarily have a low cross-correlation spectrum. Hence, for various system loads there may be users that experience high probability of error due to high cross-correlation of their assigned signature to others in the set. In this paper we present a search method to obtain new optimum TSC binary signature sets with low cross-correlation spectrum. We also show, based on our search method, that if we relax the TSC optimality constraint, codes with better cross-correlation spectrum may be found. In addition, a catastrophic binary code (i.e. a code having complementary codewords) can be TSC-optimal. Therefore, practically speaking, it is important that the TSC-optimality be sought along with the minimization of the cross-correlation spectrum. Finally our codes are tested on fully-loaded and overloaded synchronous CDMA systems, and simulation results show that they are a promising alternative to the Hadamard-Walsh codes, since they offer higher capacity.