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

The first lower bound on the peak-to-average power ratio (PAPR) of a consta nt energy code of a given length n, minimum Euclidean distance and rate is established. Conversely, using a nonconstructive Varshamov-Gilbert style ar gument yields a lower bound on the achievable rate of a code of a given len gth, minimum Euclidean distance and maximum PAPR, The derivation of these b ounds 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 log n, These bounds motivate the explicit construction of error-corr ecting codes with low PAPR, Bounds for exponential sums over Galois fields and rings are applied to obtain an upper bound of order (log n)(2) on the P APRs of a constructive class of codes, the trace codes. This class includes the binary simplex code, duals of binary, primitive Bose-Chaudhuri-Hocquen ghem (BCH) codes and a variety of their nonbinary analogs. Some open proble ms are identified.