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In this paper, we introduce a new class of codes for overloaded synchronous wireless and optical code-division multiple-access (CDMA) systems which increases the number of users for fixed number of chips without introducing any errors. Equivalently, the chip rate can be reduced for a given number of users, which implies bandwidth reduction for downlink wireless systems. An upper bound for the maximum number of users for a given number of chips is derived. Also, lower and upper bounds for the sum channel capacity of a binary overloaded CDMA are derived that can predict the existence of such overloaded codes. We also propose a simplified maximum likelihood method for decoding these types of overloaded codes. Although a high percentage of the overloading factor degrades the system performance in noisy channels, simulation results show that this degradation is not significant. More importantly, for moderate values of E b/N0 (in the range of 6-10 dB) or higher, the proposed codes perform much better than the binary Welch bound equality sequences.