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Axel Thue's work on repetitions in words
 Invited Lecture at the 4th Conference on Formal Power Series and Algebraic Combinatorics
, 1992
"... The purpose of this survey is to present, in contemporary terminology, the fundamental contributions of Axel Thue to the study of combinatorial properties of sequences of symbols, insofar as repetitions are concerned. The present state of the art is also sketched. ..."
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Cited by 22 (3 self)
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The purpose of this survey is to present, in contemporary terminology, the fundamental contributions of Axel Thue to the study of combinatorial properties of sequences of symbols, insofar as repetitions are concerned. The present state of the art is also sketched.
On cosets of the generalized firstorder ReedMuller code with low PMEPR
 IEEE Trans. Inform. Theory
, 2006
"... Golay sequences are well suited for the 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 2 m organiz ..."
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Cited by 14 (3 self)
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Golay sequences are well suited for the 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 2 m 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.
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 13 (7 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.
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 10 (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.
Aperiodic Univariate and Multivariate Merit Factors
 SETA’04, Sequences and their Applications, Seoul, Accepted for Proceedings of SETA04, Lecture Notes in Computer Science
, 2004
"... Abstract. Merit factor of a binary sequence is reviewed, and constructions are described that appear to satisfy an asymptotic merit factor of 6.3421... Multivariate merit factor is characterised and recursive Boolean constructions are presented which satisfy a nonvanishing asymptote in multivariate ..."
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Cited by 6 (5 self)
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Abstract. Merit factor of a binary sequence is reviewed, and constructions are described that appear to satisfy an asymptotic merit factor of 6.3421... Multivariate merit factor is characterised and recursive Boolean constructions are presented which satisfy a nonvanishing asymptote in multivariate merit factor. Clifford merit factor is characterised as a generalisation of multivariate merit factor and as a type of quantum merit factor. Recursive Boolean constructions are presented which, however, only satisfy an asymptotic Clifford merit factor of zero. It is demonstrated that Boolean functions obtained via quantum error correcting codes tend to maximise Clifford merit factor. Results are presented as to the distribution of the above merit factors over the set of binary sequences and Boolean functions. 1
The multivariate merit factor of a Boolean function
 PROC. IEEE INFORMATION THEORY WORKSHOP ON CODING AND COMPLEXITY – ITW 2005, 2005
, 2005
"... A new metric, the multivariate merit factor (MMF) of a Boolean function, is presented, and various infinite recursive quadratic sequence constructions are given for which both univariate and multivariate merit factors can be computed exactly. In some cases these constructions lead to merit factors w ..."
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Cited by 6 (3 self)
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A new metric, the multivariate merit factor (MMF) of a Boolean function, is presented, and various infinite recursive quadratic sequence constructions are given for which both univariate and multivariate merit factors can be computed exactly. In some cases these constructions lead to merit factors with nonvanishing asymptotes. A formula for the average value of 1 is derived and a characterisation of the MMF in terms MMF of cryptographic differentials is discussed.
On the Aperiodic Autocorrelation of Binary Sequences
, 2003
"... This master thesis (hovedfag) looks at the aperiodic autocorrelation of binary sequences. We give an overview of search techniques and classes of sequences with low aperiodic autocorrelation sidelobes, and present two new classes. One of them is the extended Legendre semiconstruction that appears ..."
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Cited by 4 (1 self)
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This master thesis (hovedfag) looks at the aperiodic autocorrelation of binary sequences. We give an overview of search techniques and classes of sequences with low aperiodic autocorrelation sidelobes, and present two new classes. One of them is the extended Legendre semiconstruction that appears to have a MF? 6:3 for large lengths. We also look at the multidimensional aperiodic autocorrelation, and present a construction for a new class of sequences with a very low multidimensional aperiodic autocorrelation.
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
Mean asymptotic behaviour of radixrational sequences and dilation equations (Extended version
, 2008
"... Abstract. The generating series of a radixrational sequence is a rational formal power series from formal language theory viewed through a fixed radix numeration system. For each radixrational sequence with complex values we provide an asymptotic expansion for the sequence of its Cesàro means. The ..."
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Cited by 2 (0 self)
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Abstract. The generating series of a radixrational sequence is a rational formal power series from formal language theory viewed through a fixed radix numeration system. For each radixrational sequence with complex values we provide an asymptotic expansion for the sequence of its Cesàro means. The precision of the asymptotic depends on the joint spectral radius of the linear representation of the sequence, and the coefficients are obtained through some dilation equations. The proofs are based on elementary linear algebra. Contents