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On the Existence and Construction Of Good Codes with Low Peak-to-Average Power Ratios

Paterson, Kenneth G.; Tarokh, Vahid

HPL-1999-51

Keyword(s): OFDM; multicarrier; power; PAPR; PMPR; PMEPR; bounds; Varshamov; Gilbert; simplex code; dual BCH code; Kerdock code; Delsarte-Goethals code; exponential sum; Lagrange interpolation; finite field; Galois ring

Abstract: The first 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 log n. These bounds motivate the explicit construction of error-correcting 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 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 identified. Notes: Vahid Tarokh, AT & T Labs - Research , 180 Park Avenue, Florham Park, New Jersey 07932, USA

22 Pages

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