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We consider the digital fingerprinting (FP) problem and model it as a multiuser communications problem and develop a detection theoretic framework. In the general case, colluders apply uniform linear averaging followed by additive colored Gaussian noise. For each user, the receiver computes the correlation between the attacked signal and a linear-transformed version of that user's fingerprint, and performs thresholding (focused detection). Assuming independent colluders with potentially unequal priors, we derive generic exact bit-error probability (BEP) expressions, together with tight bounds, for arbitrary FP codes. Then, we specialize our results to orthogonal, simplex and Gaussian codes in the presence of additive white Gaussian noise; under mean squared error distortion constraints on the embedder and the colluders, we analytically quantify the optimal detection rule, the resulting minimum BEP and its asymptotic behavior, the collusion resistance, and the error exponent for the aforementioned codes, and compare their performances. We show that the minimum BEP expressions for these codes obey the same functional form and that they can be ordered as simplex, orthogonal, and Gaussian in terms of increasing BEP.