Generalized Reed-Muller codes and power control in OFDM modulation

Controlling the peak-to-mean envelope power ratio (PMEPR) of orthogonal fre quency-division multiplexed (OFDM) transmissions is a notoriously difficult problem, though one which is of vital importance for the practical applica tion of OFDM in low-cost applications. The utility of Golay complementary s equences in solving this problem has been recognized for some time. In this paper, a powerful theory linking Golay complementary sets of polyphase seq uences and Reed-Muller codes is developed, Our main result shows that any s econd-order coset of a q-ary generalization of the first order Reed-Muller code can be partitioned into Golay complementary sets whose size depends on ly on a single parameter that is easily computed from a graph associated wi th the coset, As a first consequence, recent results of Davis and Jedwab on Golay pairs, as well as earlier constructions of Golay, Budisin and Sivasw amy are shown to arise as special cases of a unified theory for Golay compl ementary sets, As a second consequence, the main result directly yields bou nds on the PMEPR's of codes formed from selected cosets of the generalized first order Reed-Muller code. These codes enjoy efficient encoding, good er ror-correcting capability, and tightly controlled PMEPR, and significantly extend the range of coding options for applications of OFDM using small num bers of carriers.