Generalized Reed-Muller codes and power control in OFDM modulation

Controlling the peak-to-mean envelope power ratio (PMEPR) of orthogonal frequency-division multiplexed (OFDM) transmissions is a notoriously difficult problem, though one which is of vital importance for the practical application of OFDM in low-cost applications. The utility of Golay complementary sequences in solving this problem has been recognized for some time. In this paper, a powerful theory linking Golay complementary sets of polyphase sequences and Reed-Muller codes is developed. Our main result shows that any second-order coset of a q-ary generalization of the first order Reed-Muller code can be partitioned into Golay complementary sets whose size depends only on a single parameter that is easily computed from a graph associated with the coset. As a first consequence, recent results of Davis and Jedwab (see Electron. Lett., vol.33, p.267-8, 1997) on Golay pairs, as well as earlier constructions of Golay (1949, 1951, 1961), Budisin (1990) and Sivaswamy (1978) are shown to arise as special cases of a unified theory for Golay complementary sets. As a second consequence, the main result directly yields bounds on the PMEPRs of codes formed from selected cosets of the generalized first order Reed-Muller code. These codes enjoy efficient encoding, good error-correcting capability, and tightly controlled PMEPR, and significantly extend the range of coding options for applications of OFDM using small numbers of carriers.