On the existence and construction of good codes with low peak-to-average power ratios

The rst lower bound on the peak-to-average power ratio (PAPR) of a constant energy code of a given length n, minimum Euclidean distance and rate is established. Conversely, using a non-constructive Varshamov-Gilbert style argument yields a lower bound on the achievable rate of a code of a given length, minimum Euclidean distance and maximum PAPR. The derivation of these bounds relies on a geometrical analysis of the PAPR of such a code. Further analysis shows that there exist asymptotically good codes whose PAPR is at most 8 logn. These bounds motivate the explicit construction of error-correcting codes with low PAPR. Bounds for exponential sums over Galois elds and rings are applied to obtain an upper bound of order (logn) on the PAPRs of a constructive class of codes, the trace codes. This class includes the binary simplex code, duals of binary, primitive BCH codes and a variety of their non-binary analogues. Some open problems are identi ed.