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Generalised Reed-Muller Codes and Power Control in OFDM Modulation

Paterson, Kenneth

HPL-98-57R1

Keyword(s): OFDM; multicarrier; encoding; power; PMEPR; Golay; complementary; sequence; set; Reed-Muller; code

Abstract: 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 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 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 on Golay pairs, as well as earlier constructions of Golay, Budisin and Sivaswamy 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 generalised 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.

30 Pages

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