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20
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.
Golay complementary array pairs
 Designs, Codes and Cryptography
"... Constructions and nonexistence conditions for multidimensional Golay complementary array pairs are reviewed. A construction for a ddimensional Golay array pair from a (d + 1)dimensional Golay array pair is given. This is used to explain and expand previously known constructive and nonexistence re ..."
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Cited by 7 (4 self)
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Constructions and nonexistence conditions for multidimensional Golay complementary array pairs are reviewed. A construction for a ddimensional Golay array pair from a (d + 1)dimensional Golay array pair is given. This is used to explain and expand previously known constructive and nonexistence results in the binary case.
AMBIGUITY FUNCTION AND FRAME THEORETIC PROPERTIES OF PERIODIC ZERO AUTOCORRELATION WAVEFORMS
"... Constant Amplitude Zero Autocorrelation (CAZAC) waveforms u are analyzed in terms of the ambiguity function Au. Elementary number theoretic considerations illustrate that peaks in Au are not stable under small pertubations in its domain. Further, it is proved that the analysis of vectorvalued CAZA ..."
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Cited by 5 (2 self)
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Constant Amplitude Zero Autocorrelation (CAZAC) waveforms u are analyzed in terms of the ambiguity function Au. Elementary number theoretic considerations illustrate that peaks in Au are not stable under small pertubations in its domain. Further, it is proved that the analysis of vectorvalued CAZAC waveforms depends on methods from the theory of frames. Finally, techniques are introduced to characterize the structure of Au, to compute u in terms of Au, and to evaluate MSE for CAZAC waveforms.
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.
Barker sequences and flat polynomials
, 2008
"... Abstract. A Barker sequence is a finite sequence of integers, each ±1, whose aperiodic autocorrelations are all as small as possible. It is widely conjectured that only finitely many Barker sequences exist. We describe connections between Barker sequences and several problems in analysis regarding t ..."
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Abstract. A Barker sequence is a finite sequence of integers, each ±1, whose aperiodic autocorrelations are all as small as possible. It is widely conjectured that only finitely many Barker sequences exist. We describe connections between Barker sequences and several problems in analysis regarding the existence of polynomials with ±1 coefficients that remain flat over the unit circle according to some criterion. First, we amend an argument of Saffari to show that a polynomial constructed from a Barker sequence remains within a constant factor of its L2 norm over the unit circle, in connection with a problem of Littlewood. Second, we show that a Barker sequence produces a polynomial with very large Mahler’s measure, in connection with a question of Mahler. Third, we optimize an argument of Newman to prove that any polynomial with ±1 coefficients and positive degree n − 1 has L1 norm less than √ n −.09, and note that a slightly stronger statement would imply that long Barker sequences do not exist. We also record polynomials with ±1 coefficients having maximal L1 norm or maximal Mahler’s measure for each fixed degree up to 24. Finally, we show that if one could establish that the polynomials in a particular sequence are all irreducible over Q, then an alternative proof that there are no long Barker sequences with odd length would follow. 1.
Probabilistic and Constructive Methods in Harmonic Analysis and Additive Number Theory
, 1994
"... We give several applications of the probabilistic method in harmonic analysis and additive number theory. We also give efficient constructions in place of previous probabilistic (existential) proofs. 1. Using the probabilistic method we prove that there exist integers p 1 ; : : : ; p N 0 for which ..."
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Cited by 1 (1 self)
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We give several applications of the probabilistic method in harmonic analysis and additive number theory. We also give efficient constructions in place of previous probabilistic (existential) proofs. 1. Using the probabilistic method we prove that there exist integers p 1 ; : : : ; p N 0 for which fi fi fi fi fi fi min x N X j=1 p j cos jx fi fi fi fi fi fi = O(s 1=3 ); as s !1, where s = P N j=1 p j . This improves a result of Odlyzko who proved a similar inequality with the right hand side replaced by O((s log s) 1=3 ). 2. Similarly we prove that there are frequencies 1 ! \Delta \Delta \Delta ! N 2 f1; : : : ; cNg, for c = 2, for which fi fi fi fi fi fi min x N X j=1 cos j x fi fi fi fi fi fi = O(N 1=2 ) and that this is impossible for smaller values of the positive constant c. 3. The previous result is used to prove easily a theorem of Erdos and Tur'an about the density of finite integer sequences with the property that any two elements have a different sum...
The Sidon constant of sets with three elements. arxiv.org/ math/0102145
, 2001
"... We solve an elementary minimax problem and obtain the exact value of the Sidon constant for sets with three elements {n0, n1, n2}: it is sec(π/2n) for n = max ni − nj/gcd(n1 − n0, n2 − n0). 1 ..."
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Cited by 1 (0 self)
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We solve an elementary minimax problem and obtain the exact value of the Sidon constant for sets with three elements {n0, n1, n2}: it is sec(π/2n) for n = max ni − nj/gcd(n1 − n0, n2 − n0). 1
EVEN MOMENTS OF GENERALIZED RUDIN–SHAPIRO POLYNOMIALS
"... Abstract. We know from Littlewood (1968) that the moments of order 4 of the classical Rudin–Shapiro polynomials Pn(z) satisfy a linear recurrence of degree 2. In a previous article, we developed a new approach, which enables us to compute exactly all the moments Mq(Pn) ofevenorderqfor q � 32. We wer ..."
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Abstract. We know from Littlewood (1968) that the moments of order 4 of the classical Rudin–Shapiro polynomials Pn(z) satisfy a linear recurrence of degree 2. In a previous article, we developed a new approach, which enables us to compute exactly all the moments Mq(Pn) ofevenorderqfor q � 32. We were also able to check a conjecture on the asymptotic behavior of Mq(Pn), namely Mq(Pn) ∼ Cq2nq/2, where Cq = 2q/2 /(q/2 + 1), for q even and q � 52. Now for every integer ℓ � 2 there exists a sequence of generalized Rudin–Shapiro polynomials, denoted by P (ℓ) (z). In this paper, we extend our earlier method to these polynomials. In particular, the moments Mq(P (ℓ) 0,n) have been completely determined for ℓ = 3 and q = 4, 6, 8, 10, for ℓ = 4 and q =4, 6andforℓ =5, 6, 7, 8andq = 4. For higher values of ℓ and q, we formulate a natural conjecture, which implies that Mq(P (ℓ) 0,n) ∼ Cℓ,qℓ nq/2, where Cℓ,q is an explicit constant. 1.