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Upper and lower bounds on the average probability of error of direct sequence (DS) spread-spectrum communication systems operating in the presence of either multiple tone or multiple access interference are presented. The bounds are quite tight and can be used without the restriction that the peak level of the interference be less than the desired signal level. A simple method for evaluating approximations to the bounds for multiple access systems, in which the peak interference does indeed exceed the desired signal level, is presented as well. The tightness of both the bounds and the approximations are illustrated with numerical results.