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 di cult 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 recognised 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 generalisation of the rst 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 rst consequence, recent results of Davis and Jedwab on Golay pairs, as well as earlier constructions of Golay, Budi sin and Sivaswamy are shown to arise as special cases of a uni ed 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 generalised rst order Reed-Muller code. These codes enjoy e cient encoding, good error correcting capability and tightly controlled PMEPR, and signi cantly extend the range of coding options for applications of OFDM using small numbers of carriers.