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Doppler resilient Golay complementary pairs for radar,” presented at the
 IEEE Statist. Signal Process. Workshop (SSP
, 2007
"... We present a systematic way of constructing a Doppler resilient sequence of Golay complementary waveforms for radar, for which the composite ambiguity function maintains ideal shape at small Doppler shifts. The idea is to determine a sequence of Golay pairs that annihilates the loworder terms of th ..."
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Cited by 17 (1 self)
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We present a systematic way of constructing a Doppler resilient sequence of Golay complementary waveforms for radar, for which the composite ambiguity function maintains ideal shape at small Doppler shifts. The idea is to determine a sequence of Golay pairs that annihilates the loworder terms of the Taylor expansion of the composite ambiguity function. The ProuhetThueMorse sequence plays a key role in the construction of Doppler resilient sequences of Golay pairs. We extend this construction to multiple dimensions. In particular, we consider radar polarimetry, where the dimensions are realized by two orthogonal polarizations. We determine a sequence of twobytwo Alamouti matrices, where the entries involve Golay pairs and for which the matrixvalued composite ambiguity function vanishes at small Doppler shifts. 1.
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
Complementary Sets, Generalized ReedMuller Codes, and Power Control for OFDM
 IEEE Trans. Inform. Theory
, 2007
"... The use of errorcorrecting codes for tight control of the peaktomean envelope power ratio (PMEPR) in orthogonal frequencydivision multiplexing (OFDM) transmission is considered in this correspondence. By generalizing a result by Paterson, it is shown that each qphase (q is even) sequence of len ..."
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Cited by 10 (1 self)
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The use of errorcorrecting codes for tight control of the peaktomean envelope power ratio (PMEPR) in orthogonal frequencydivision multiplexing (OFDM) transmission is considered in this correspondence. By generalizing a result by Paterson, it is shown that each qphase (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 qphase sequences of length 2m. A new 2hary 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.
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.
New Constructions of General QAM Golay Complementary Sequences
, 2012
"... There have been five constructions (Case I to Case V) of 64QAM Golay complementary sequences (GCSs), of which the Case IV and Case V constructions were identified by Chang, Li, and Hirata in 2010. The Generalized Cases IIII constructions for 4qQAM (q ≥ 1) GCSs were additionally proposed by Li. In ..."
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Cited by 6 (2 self)
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There have been five constructions (Case I to Case V) of 64QAM Golay complementary sequences (GCSs), of which the Case IV and Case V constructions were identified by Chang, Li, and Hirata in 2010. The Generalized Cases IIII constructions for 4qQAM (q ≥ 1) GCSs were additionally proposed by Li. In this paper, the Generalized Case IV and Generalized Case V constructions for 4qQAM (q> = 3) GCSs are proposed using selected Gaussian integer pairs, each of which contains two distinct Gaussian integers with identical magnitude and which are not conjugate with each other.
The PAPR Problem in OFDM Transmission: New Directions for a LongLasting Problem
 IEEE SIGNAL PROCESSING MAGAZINE
, 2012
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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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Cited by 3 (2 self)
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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
A Database for Boolean Functions and Constructions of . . .
, 2008
"... In this thesis, we study spectral measures of Boolean functions. In the first half of thesis, we study the Walsh spectrum and the periodic autocorrelation spectrum of a Boolean function. We give a survey on the cryptographic criteria implemented in the developed Boolean function database. In the s ..."
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Cited by 2 (0 self)
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In this thesis, we study spectral measures of Boolean functions. In the first half of thesis, we study the Walsh spectrum and the periodic autocorrelation spectrum of a Boolean function. We give a survey on the cryptographic criteria implemented in the developed Boolean function database. In the second half of the thesis, we study the aperiodic autocorrelation spectrum of a Boolean function and some more spectral measures with respect to certain types of unitary matrices. We investigate the Turyn construction for Golay complementary pairs. We show how to convert this construction so as to realize three distinct types of complementary construction. We focus, in particular, on the construction of Boolean function pairs which are TypeI, TypeII or TypeIII complementary or nearcomplementary. iii Acknowledgements First and foremost, I would like to thank my supervisor M. Parker for showing me how scientific research is done. Thank you for showing me how theorems and lemmas are