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The paper considers a coding scheme for multipath channel estimation in uplink code-division multiple-access systems where each user transmits an individual pilot signal (sequence) to estimate its multipath channel coefficients. Assuming a common radio channel model with a uniform power delay profile, we derive lower bounds on the maximum mean square error for two types of linear channel estimators: an inverse filter and a linear minimum mean square error (MMSE) estimator. In contrast to previous work, the main focus here is on overloaded systems where the total number of multipath channel coefficients of all users is greater than processing gain. We show that the inverse filter bound is attained if and only if each pilot sequence is a perfect root-of-unity sequence. Interestingly, the conventional matched filter achieves the same lower bound if pilot sequences form a complementary periodic sequence set. In case of the MMSE estimator, the lower bound is either met or not depending on some system parameters. We provide a necessary and sufficient condition for achieving the bound when pilot sequences are arbitrary vectors on the unit sphere. This paper gives insight into the performance limits of practical systems.