On Cosets of the Generalized First-Order Reed-Muller Code with Low PMEPR
  Golay sequences are well suited for the use as codewords in orthogonal frequency-division multiplexing (OFDM), since their peak-to-mean envelope power ratio (PMEPR) in q-ary phase-shift keying (PSK) modulation is at most 2. It is known that a family of polyphase Golay sequences of length 2^m organizes in m!/2 cosets of a q-ary generalization of the first-order Reed-Muller code, RM_q(1,m). In this paper a more general construction technique for cosets of RM_q(1,m) with low PMEPR is established. These cosets contain so-called near-complementary sequences. The application of this theory is then illustrated by providing some construction examples. First, it is shown that the m!/2 cosets of RM_q(1,m) comprised of Golay sequences just arise as a special case. Second, further families of cosets of RM_q(1,m) with maximum PMEPR between 2 and 4 are presented, showing that some previously unexplained phenomena can now be understood within a unified framework. A lower bound on the PMEPR of cosets of RM_q(1,m) is proved as well, and it is demonstrated that the upper bound on the PMEPR is tight in many cases. Finally it is shown that all upper bounds on the PMEPR of cosets of RM_q(1,m) also hold for the peak-to-average power ratio (PAPR) under the Walsh-Hadamard transform. 
Kai-Uwe Schmidt