On codes with low peak-to-average power ratio for multi-code CDMA

Abstract

Codes which reduce the peak-to-average power (PAPR) in multi-code code division multiple access (MC-CDMA) communications systems are systematically studied. The problem of designing such codes is reformulated as a new coding-theoretic problem: codes with low PAPR are ones in which the codewords are far from the first-order ReedMuller code. Bounds on the trade-off between rate, PAPR and error-correcting capability of codes for MC-CDMA follow. The connections between the code design problem, bent functions and algebraic coding theory (in particular, the Kerdock codes and Delsarte-Goethals codes) are exploited to construct code families with flexible parameters for the small values of n of practical interest. In view of their algebraic structure, these codes enjoy efficient encoding and decoding algorithms. The paper concludes by listing open problems in algebraic coding theory and Boolean functions motivated by the paper.

Peak-to-average power ratio analysis in multicode CDMA

“…In fact, some probabilities can be computed exactly. Paterson has shown that for any word c of length n [18], …”

“…In [16,17] the impact of an amplifier nonlinearity is investigated for various multicode systems. In [18,19], PAPR reduction codes are systematically studied. Codes which are distant(Hamming sense) from the first-order Reed-Muller code will have small PAPR.…”

Galois Rings and Pseudo-random Sequences

“…involving not only additive but also multiplicative characters) in relation to many valued sequences [17]. In the second approach, motivated by multi-code CDMA, we are to control the nonlinearity of binary sequences of length a power of 2 [16]. This notion of nonlinearity coincides with the one familiar from boolean functions theory.…”

“…In both models sequences live in families and are supposed to form a code either spherical in the first case or binary in the second. Both papers [17,16] already use weighted degree trace codes and our contribution is mostly of extension from Z 4 to Z 2 for > 2 [22], or to non primitive periods [24]. In general, the use of larger rings provides extra flexibility in the design choice; the sequences families constructed in that way having a larger size but a somewhat larger PAPR.…”

“…Section 5 considers the binary sequence constructed from sequences over Z 8 that was introduced in [12]. Section 6 considers short binary sequences of [22], motivated by [16]. Section 7 contains the binary sequences constructed by taking the MSB of 2 l −ary sequences [21].Section 8 contains the PAPR reduction sequences over large rings of [24].…”