The peak-to-average power ratio PAPR(C) of a code C is an important characteristic of that code when it is used in OFDM communications. We establish bounds on the region of achievable triples (R, d, PAPR(C)) where R is the code rate and d is the minimum Euclidean distance of the code. We prove a lower bound on PAPR in terms of R and d and show that there exist asymptotically good codes whose PAPR is at most 8 log n. We give explicit constructions of error-correcting codes with low PAPR by employing bounds for hybrid exponential sums over Galois fields and rings.