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A constant-amplitude code is a code that reduces the peak-to-average power ratio (PAPR) in multicode code-division multiple access (MC-CDMA) systems to the favorable value 1. In this paper, quaternary constant-amplitude codes (codes over [Z.sub.4]) of length [2.sup.m] with error-correction capabilities are studied. These codes exist for every positive integer m, while binary constant-amplitude codes cannot exist if m is odd. Every word of such a code corresponds to a function from the binary m-tuples to [Z.sup.4] having the bent property, i.e., its Fourier transform has magnitudes [2.sup.m/2]. Several constructions of such functions are presented, which are exploited in connection with algebraic codes over [Z.sub.4] (in particular quaternary Reed-Muller, Kerdock, and Delsarte-Goethals codes) to construct families of quaternary constant-amplitude codes. Mappings from binary to quaternary constant-amplitude codes are presented as well. Index Terms--Bent function, code, code-division multiple access (CDMA), Delsarte-Goethals, Kerdock, multicode, peak-to-average power ratio (PAPR), quaternary, Reed-Muller.

The peak-to-mean envelope power ratio (PMEPR) of a code employed in orthogonal frequency-division multiplexing (OFDM) systems can be reduced by permuting its coordinates and by rotating each coordinate by a fixed phase shift. Motivated by some previous designs of phase shifts using suboptimal methods, the following question is considered in this paper. For a given binary code, how much PMEPR reduction can be achieved when the phase shifts are taken from a [2.sup.h]-ary phase-shift keying ([2.sup.h]-PSK) constellation? A lower bound on the achievable PMEPR is established, which is related to the covering radius of the binary code. Generally speaking, the achievable region of the PMEPR shrinks as the covering radius of the binary code decreases. The bound is then applied to some well understood codes, including nonredundant BPSK signaling, BCH codes and their duals, Reed-Muller codes, and convolutional codes. It is demonstrated that most (presumably not optimal) phase-shift designs from the literature attain or approach our bound. Index Terms--BCH codes, convolutional codes, covering radius, orthogonal frequency-division multiplexing (OFDM), peak-to-average power ratio (PAPR), peak-to-mean envelope power ratio (PMEPR), Reed-Muller codes.

The use of error-correcting codes for tight control of the peak-to-mean envelope power ratio (PMEPR) in orthogonal frequency-division multiplexing (OFDM) transmission is considered in this correspondence. By generalizing a result by Paterson, it is shown that each q-phase (q is even) sequence of length [2.sup.m] lies in a complementary set of size [2.sup.k+1], where k is a nonnegative integer that can be easily determined from the generalized Boolean function associated with the sequence. For small k this result provides a reasonably tight bound for the PMEPR of q-phase sequences of length [2.sup.m]. A new [2.sup.h]-ary generalization of the classical Reed--Muller code is then used together with the result on complementary sets to derive flexible OFDM coding schemes with low PMEPR. These codes include the codes developed by Davis and Jedwab as a special case. In certain situations the codes in the present correspondence are similar to Paterson's code constructions and often outperform them. Index Terms--Code, complementary, correlation, Golay, orthogonal frequency-division multiplexing (OFDM), peak-to-mean envelope power ratio (PMEPR), Reed--Muller, sequence, set.