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 m lies in a complementary set of size 2 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 m . A new 2 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.

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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 4 ) of length 2 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 4 having the bent property, i.e., its Fourier transform has magnitudes 2 m/2 . Several constructions of such functions are presented, which are exploited in connection with algebraic codes over Z 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 TermsBent function, code, code-division multiple access (CDMA), Delsarte-Goethals, Kerdock, multicode, peak-to-average power ratio (PAPR), quaternary, Reed-Muller of analog devices in the transmission chain, such as the power amplifier, digital-to-analog, and analog-to-digital converters, is limited due to the high PAPR of the signals.An elegant solution to solve this power-control problem is to draw the modulating words from a block code that contains only words with low PAPR and, simultaneously, has error-correction capabilities. This idea was originally proposed for orthogonal frequency-division multiplexing (OFDM) systems [9], where a similar power-control problem occurs. We will see that the PAPR is always at least 1. A code for which all codewords achieve this lower bound is called a constant-amplitude code.A coding-theoretic framework for binary codes in MC-CDMA has been established by Paterson [17]. It was shown that codewords with low PAPR are exactly those words that are far from the first-order Reed-Muller code, RM(1, m). This fact was used in [17] to prove fundamental bounds on the trade-off between PAPR, minimum distance, and rate of binary codes. Moreover several families of binary constant-amplitude codes have been constructed in [17] by exploiting the relation between bent functions [19], [13] and binary constant-amplitude codewords (a connection that was first recognized by Wada [24]). These codes are unions of cosets of RM(1, m) lying in higher-order Reed-Muller, Kerdock, or Delsarte-Goethals codes. Therefore they enjoy high minimum distance and are amenable to efficient encoding and decoding algorithms.Solé and Zinoviev [23] constructed binary codes with PAPR much greater than 1, which makes them less attractive for practical values of n. However in many situations their parameters asymptotically beat the Gilbert-Varshamov-style lower bound derived in [17].While previous work was focused on binary codes, several motivations exist to study quaternary codes for MC-CDMA. First, quaternary modulation rather than binary modulation is often employed in MC-CDMA systems [4]. Second, binary const...

Abstract. Classical planar functions are functions from a finite field to itself and give rise to finite projective planes. They exist however only for fields of odd characteristic. We study their natural counterparts in characteristic two, which we also call planar functions. They again give rise to finite projective planes, as recently shown by the second author. We give a characterisation of planar functions in characteristic two in terms of codes over Z4. We then specialise to planar monomial functions f (x) = cx t and present constructions and partial results towards their classification. In particular, we show that t = 1 is the only odd exponent for which f (x) = cx t is planar (for some nonzero c) over infinitely many fields. The proof techniques involve methods from algebraic geometry.