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12
On Cosets of the Generalized FirstOrder Reed–Muller Code with Low PMEPR
, 2006
"... Golay sequences are well suited for use as codewords in orthogonal frequencydivision multiplexing (OFDM) since their peaktomean envelope power ratio (PMEPR) in qary phaseshift keying (PSK) modulation is at most 2. It is known that a family of polyphase Golay sequences of length 2m organizes in ..."
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Cited by 23 (3 self)
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Golay sequences are well suited for use as codewords in orthogonal frequencydivision multiplexing (OFDM) since their peaktomean envelope power ratio (PMEPR) in qary phaseshift keying (PSK) modulation is at most 2. It is known that a family of polyphase Golay sequences of length 2m organizes in m!/2 cosets of a qary generalization of the firstorder Reed–Muller code, RMq(1, m). In this paper a more general construction technique for cosets of RMq(1, m) with low PMEPR is established. These cosets contain socalled nearcomplementary sequences. The application of this theory is then illustrated by providing some construction examples. First, it is shown that the m!/2 cosets of RMq(1, m) comprised of Golay sequences just arise as a special case. Second, further families of cosets of RMq(1, m) with maximum PMEPR between 2 and 4 are presented, showing that some previously unexplained phenomena can now be understood within a unified framework. A lower bound on the PMEPR of cosets of RMq(1, m) is proved as well, and it is demonstrated that the upper bound on the PMEPR is tight in many cases. Finally it is shown that all upper bounds on the PMEPR of cosets of RMq(1, m) also hold for the peaktoaverage power ratio (PAPR) under the Walsh–Hadamard transform.
The Quantum Entanglement of Binary and Bipolar Sequences
 in Sequences and Their Applications, Discrete Mathematics and Theoretical Computer Science Series
, 2001
"... This paper highlights two partial entanglement measures, namely the 'Linear Entanglement' (LE) (Section 6, Definition 11), and 'Stubborness of Entanglement' (SE) (Section 7, Definition 16), which is a sequence of parameters. The paper is aimed at both coding theorists and sequenc ..."
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Cited by 17 (12 self)
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This paper highlights two partial entanglement measures, namely the 'Linear Entanglement' (LE) (Section 6, Definition 11), and 'Stubborness of Entanglement' (SE) (Section 7, Definition 16), which is a sequence of parameters. The paper is aimed at both coding theorists and sequence designers, and at quantum physicists, and argues that the best codes and/or sequences can be interpreted as describing multiparticle states with high entanglement. A binary linear errorcorrecting code (ECC), C, is often partially described by its parameters [n, k, d], where n is wordlength, k is code dimension, and d is minimum Hamming Distance [18], and more generally by its weight hierarchy. We show, by interpreting the length 2 n indicator for C as an nparticle quantum state that, for those states representing binary linear ECCs, the ECCs with optimal weight hierarchy also have optimal LE and optimal SE (Theorems 10 and 15). By action of local unitary transform on the indicator of C, we can also view the quantum state as a bipolar sequence. In this context a sequence is often partially described by its nonlinear order, N , and correlation immunity order, CI (Definitions 21, 22). We show that N and CI give a lower bound on 2 Matthew G. Parker and V. Rijmen LE (Theorem 16). LE is the n  log 2 of a spectral 'peak' measure of Peakto Average Power Ratio (PAR l (Section 6, Definition 10)), which is also an important measure in telecommunications [7,21,20]. This paper refers both to PAR l and to LE, where the two parameters are trivially related (Definition 11). The quantummechanical rule of 'local unitary equivalence' is a generalisation of code duality. We now state the most important results of this paper. We emphasise quantum states s from the set # p , where # p is equivalent to t...
Generalised RudinShapiro Constructions
 WCC2001, WORKSHOP ON CODING AND CRYPTOGRAPHY, PARIS(FRANCE
, 2001
"... A Golay Complementary Sequence (CS) has PeaktoAveragePowerRatio (PAPR) ≤ 2.0 for its onedimensional continuous Discrete Fourier Transform (DFT) spectrum. Davis and Jedwab showed that all known length 2 m CS, (GDJ CS), originate from certain quadratic cosets of ReedMuller (1, m). These can be g ..."
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Cited by 15 (8 self)
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A Golay Complementary Sequence (CS) has PeaktoAveragePowerRatio (PAPR) ≤ 2.0 for its onedimensional continuous Discrete Fourier Transform (DFT) spectrum. Davis and Jedwab showed that all known length 2 m CS, (GDJ CS), originate from certain quadratic cosets of ReedMuller (1, m). These can be generated using the RudinShapiro construction. This paper shows that GDJ CS have PAPR ≤ 2.0 under all unitary transforms whose rows are unimodular linear (Linear Unimodular Unitary Transforms (LUUTs)), including one and multidimensional generalised DFTs. We also propose tensor cosets of GDJ sequences arising from RudinShapiro extensions of nearcomplementary pairs, thereby generating many infinite sequence families with tight low PAPR bounds under LUUTs.
A Construction for Binary Sequence Sets with Low PeaktoAverage Power Ratio
"... A recursive construction is provided for sequence sets which possess good Hamming Distance and low PeaktoAverage Power Ratio (PAR) under any Local Unitary Unimodular Transform (including all one and multidimensional Discrete Fourier Transforms). An important instance of the construction identifie ..."
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Cited by 14 (9 self)
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A recursive construction is provided for sequence sets which possess good Hamming Distance and low PeaktoAverage Power Ratio (PAR) under any Local Unitary Unimodular Transform (including all one and multidimensional Discrete Fourier Transforms). An important instance of the construction identifies an iteration and specialisation of the MaioranaMcFarland (MM) construction. I.
Nearcomplementary Sequences With Low PMEPR for Peak Power Control in Multicarrier Communications
, 2009
"... New families of nearcomplementary sequences are presented for peak power control in multicarrier communications. A framework for nearcomplementary sequences is given by the explicit Boolean expression and the equivalent array structure. The framework transforms the seed pairs to nearcomplementary ..."
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Cited by 3 (0 self)
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New families of nearcomplementary sequences are presented for peak power control in multicarrier communications. A framework for nearcomplementary sequences is given by the explicit Boolean expression and the equivalent array structure. The framework transforms the seed pairs to nearcomplementary sequences by the aid of Golay complementary sequences. As the first example, a new sequence family of length 2m and peaktomean envelope power ratio (PMEPR) ≤ 4 is presented, where the family produces more distinct sequences than any other known near complementary sequences of the same lengths and PMEPR bound. An efficient generation algorithm for permutations is developed for the distinct sequences. In addition, new families of nearcomplementary sequences of various lengths and PMEPR < 4 are also presented, where the sequences are constructed by the framework employing the seeds of shortened or extended Golay complementary pairs. The families present in a constructive way a large number of sequences of PMEPR < 4 for the lengths (< 100) of
The Quantum Entanglement of Bipolar Sequences
 Sequences and their Applications, SETA’01
, 2001
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Spectrally Bounded Sequences, Codes and States: Graph Constructions and Entanglement
 EIGHTH IMA INTERNATIONAL CONFERENCE ON CRYPTOGRAPHY AND CODING
, 2001
"... A recursive construction is provided for sequence sets which possess good Hamming Distance and low PeaktoAverage Power Ratio (PAR) under any Local Unitary Unimodular Transform. We identify a subset of these sequences that map to binary indicators for linear and nonlinear Factor Graphs, after appl ..."
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Cited by 1 (1 self)
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A recursive construction is provided for sequence sets which possess good Hamming Distance and low PeaktoAverage Power Ratio (PAR) under any Local Unitary Unimodular Transform. We identify a subset of these sequences that map to binary indicators for linear and nonlinear Factor Graphs, after application of subspace WalshHadamard Transforms. Finally we investigate the quantum PARl measure of ’Linear Entanglement’ (LE) under any Local Unitary Transform, where optimum LE implies optimum weight hierarchy of an associated linear code.
A Generalized Construction of OFDM MQAM Sequences With Low PeaktoAverage Power Ratio
, 2010
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