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15
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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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.
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.
A multidimensional approach to the construction and enumeration of Golay complementary sequences
 J. Combin. Theory (A
, 2006
"... We argue that a Golay complementary sequence is naturally viewed as a projection of a multidimensional Golay array. We present a threestage process for constructing and enumerating Golay array and sequence pairs: 1. construct suitable Golay array pairs from lowerdimensional Golay array pairs; 2. a ..."
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Cited by 14 (8 self)
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We argue that a Golay complementary sequence is naturally viewed as a projection of a multidimensional Golay array. We present a threestage process for constructing and enumerating Golay array and sequence pairs: 1. construct suitable Golay array pairs from lowerdimensional Golay array pairs; 2. apply transformations to these Golay array pairs to generate a larger set of Golay array pairs; and 3. take projections of the resulting Golay array pairs to lower dimensions. This process greatly simplifies previous approaches, by separating the construction of Golay arrays from the enumeration of all possible projections of these arrays to lower dimensions. We use this process to construct and enumerate all 2 hphase Golay sequences of length 2 m obtainable under any known method, including all 4phase Golay sequences obtainable from the length 16 examples given in 2005 by Li and Chu [12]. 1
GolayDavisJedwab Complementary Sequences and RudinShapiro Constructions
 IEEE TRANS. INFORM. THEORY
, 2001
"... A Golay Complementary Sequence (CS) has a 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 b ..."
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Cited by 12 (4 self)
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A Golay Complementary Sequence (CS) has a 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 a PAPR ≤ 2.0 under all 2 m ×2 m unitary transforms whose rows are unimodular linear (Linear Unimodular Unitary Transforms (LUUTs)), including one and multidimensional generalised DFTs. In this context we define Constahadamard Transforms (CHTs) and show how all LUUTs can be formed from tensor combinations of CHTs. We also propose tensor cosets of GDJ sequences arising from RudinShapiro extensions of nearcomplementary pairs, thereby generating many more infinite sequence families with tight low PAPR bounds under LUUTs. We m m−⌊ then show that GDJ CS have a PAPR ≤ 2 2 ⌋ under all 2m × 2m unitary transforms whose rows are linear (Linear Unitary Transforms (LUTs)). Finally we present a radix2 tensor decomposition of any 2 m × 2 m LUT.
A Framework for the Construction of Golay Sequences
, 2006
"... ... normal form for m! · 2 h(m+2) ordered Golay pairs of length 2 m over Z 2 h, involving m!/2 · 2 h(m+1) Golay sequences. In 2005 Li and Chu unexpectedly found an additional 1024 length 16 quaternary Golay sequences. Fiedler and Jedwab showed in 2006 that these new Golay sequences exist because of ..."
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Cited by 9 (3 self)
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... normal form for m! · 2 h(m+2) ordered Golay pairs of length 2 m over Z 2 h, involving m!/2 · 2 h(m+1) Golay sequences. In 2005 Li and Chu unexpectedly found an additional 1024 length 16 quaternary Golay sequences. Fiedler and Jedwab showed in 2006 that these new Golay sequences exist because of a “crossover” of the aperiodic autocorrelation function of certain quaternary length 8 sequences belonging to Golay pairs, and that they spawn further new quaternary Golay sequences and pairs of length 2 m for m> 4 under Budiˇsin’s 1990 iterative construction. The total number of Golay sequences and pairs spawned in this way is counted, and their algebraic normal form is given explicitly. A framework of constructions is derived in which Turyn’s 1974 product construction, together with several variations, plays a key role. All previously known Golay sequences and pairs of length 2 m over Z 2 h can be obtained directly in explicit algebraic normal form from this framework. Furthermore, additional quaternary Golay sequences and pairs of length 2 m are produced that cannot be obtained from any other known construction. The framework generalizes readily to lengths that are not a power of 2, and to alphabets other than Z 2 h.
Close encounters with Boolean functions of three different kinds
"... Complex arrays with good aperiodic properties are characterised and it is shown how the joining of dimensions can generate sequences which retain the aperiodic properties of the parent array. For the case of 2 × 2 ×... × 2 arrays we define two new notions of aperiodicity by exploiting a unitary ma ..."
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Cited by 7 (5 self)
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Complex arrays with good aperiodic properties are characterised and it is shown how the joining of dimensions can generate sequences which retain the aperiodic properties of the parent array. For the case of 2 × 2 ×... × 2 arrays we define two new notions of aperiodicity by exploiting a unitary matrix represention. In particular, we apply unitary rotations by members of a size3 cyclic subgroup of the local Clifford group to the aperiodic description. It is shown how the three notions of aperiodicity relate naturally to the autocorrelations described by the action of the HeisenbergWeyl group. Finally, after providing some cryptographic motivation for two of the three aperiodic descriptions, we devise new constructions for complementary pairs of Boolean functions of three different kinds, and give explicit examples for each.
Even Length Binary Sequence Families with Low Negaperiodic Autocorrelation
 SpringerVerlag LNCS 2227, AAECC14 Pros., pp 200210 , Nov 2001
, 2001
"... Cyclotomic constructions are given for several infinite families of even length binary sequences which have low negaperiodic autocorrelation. It appears that two of the constructions have asymptotic Merit Factor 6.0 which is very high. Mappings from periodic to negaperiodic autocorrelation are also ..."
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Cited by 3 (1 self)
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Cyclotomic constructions are given for several infinite families of even length binary sequences which have low negaperiodic autocorrelation. It appears that two of the constructions have asymptotic Merit Factor 6.0 which is very high. Mappings from periodic to negaperiodic autocorrelation are also discussed. 1
Generalised complementary arrays
 Lecture Notes in Computer Science, LNCS 7089
, 2011
"... Abstract. We present a generalised setting for the construction of complementary array pairs and its proof, using a unitary matrix notation. When the unitaries comprise multivariate polynomials in complex space, we show that four definitions of conjugation imply four types of complementary pair typ ..."
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Abstract. We present a generalised setting for the construction of complementary array pairs and its proof, using a unitary matrix notation. When the unitaries comprise multivariate polynomials in complex space, we show that four definitions of conjugation imply four types of complementary pair types I, II, III, and IV. We provide a construction for complementary pairs of types I, II, and III over {1, −1}, and further specialize to a construction for all known 2 × 2 ×... × 2 complementary array pairs of types I, II, and III over {1, −1}. We present a construction for typeIV complementary array pairs, and call them Rayleigh quotient pairs. We then generalise to complementary array sets, provide a construction for complementary sets of types I, II, and III over {1, −1}, further specialize to a construction for all known 2 × 2 ×... × 2 complementary array sets of types I, II, and III over {1, −1}, and derive closedform Boolean formulas for these cases.
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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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
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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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.