On cosets of the generalized first-order Reed--Muller code with low PMEPR

Golay sequences are well suited for 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.sup.m] organizes in m!/2 cosets of a q-ary generalization of the first-order Reed--Muller code, [RM.sub.q] (1, m). In this paper, a more general construction technique for cosets of [RM.sub.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.sub.q] (1, m) comprised of Golay sequences just arise as a special case. Second, further families of cosets of [RM.sub.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.sub.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.sub.q] (1, m) also hold for the peak-to-average power ratio (PAPR) under the Walsh--Hadamard transform (WHT). Index Terms--Aperiodic, code, complementary, correlation, Golay, orthogonal frequency-division multiplexing (OFDM), peak-to-average power ratio (PAPR), peak-to-mean envelope power ratio (PMEPR), Reed--Muller, Rudin--Shapiro, sequence, Walsh--Hadamard.