Codes, Correlations and Power Control in OFDM

power, envelope, OFDM, Reed-Muller, code, Golay, complementary, sequence, pair, set Practical communications engineering is continuously producing problems in interest to the coding theory community. A recent example is the power-control problem in Orthogonal Frequency Division Multiplexing (OFDM). We report recent work which gives a mathematical framework for generating solutions to this notorious problem that are suited to low-cost wireless applications. The key result is a connection between Golay complementary sequences and Reed-Muller codes. The former are almost ideal for OFDM transmissions because they have a very low peak-to-mean envelope power ratio (PMEPR), while the latter have efficient encoding and decoding algorithms and good error correction capability. This result is then generalised in two ways. Firstly we study polyphase Golay sequences, motivating the introduction of non-binary generalisations of the Reed-Muller codes. Secondly we consider Golay complementary sets, where the results can be presented most naturally in the language of graph theory. The practical impact is a flexible family of OFDM codes which combine low PMEPR with good error correction capability. However, the interaction between theory and practice is a two-way process: the application motivates further study of a fertile interplay between coding theory, graph theory and sequence design. We include a list of open problems which we hope will stimulate further research in this area.