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14
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.
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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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.
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.
Equivalence Between Certain Complementary Pairs of Types I and III
"... Building on a recent paper which defined complementary array pairs of types I, II, and III, this paper further characterises a class of typeI pairs defined over the alphabet {−1, 0, 1} and shows that a subset of these pairs are localunitaryequivalent to a subset of the typeIII pairs defined over ..."
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Cited by 6 (5 self)
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Building on a recent paper which defined complementary array pairs of types I, II, and III, this paper further characterises a class of typeI pairs defined over the alphabet {−1, 0, 1} and shows that a subset of these pairs are localunitaryequivalent to a subset of the typeIII pairs defined over a bipolar ({1, −1}) alphabet. Enumerations of the distinct structures in this class and its subset are given.
Polynomial Residue Systems via Unitary Transforms
"... A polynomial, A(z), can be represented by a polynomial residue system and, given enough independent residues, the polynomial can be reconstituted from its residues by the Chinese remainder theorem (CRT). A special case occurs when the discrete Fourier transform and its inverse realise the residue ev ..."
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Cited by 4 (3 self)
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A polynomial, A(z), can be represented by a polynomial residue system and, given enough independent residues, the polynomial can be reconstituted from its residues by the Chinese remainder theorem (CRT). A special case occurs when the discrete Fourier transform and its inverse realise the residue evaluations and CRT respectively, in which case the residue system is realised by the action of a matrix transform that is unitary. In this paper we generalise the class of residue systems that can be realised by the action of unitary transforms beyond the Fourier case, by suitable modification of the polynomial, A(z). We identify two new types of such system that are of particular interest, and also extend from the univariate to the multivariate case. By way of example, we show how the generalisation leads to two new types of complementary array pair. 1. Polynomial Residue Systems Let A(z) = (A0 + A1z +... + AN−1z N−1) be a univariate polynomial with coefficients A = (A0, A1,..., AN−1) ∈ C N, for C the field of complex numbers. One can embed A(z) in a polynomial modulus M(z), where A(z) = A(z) mod M(z), iff deg(M(z)) ≥ N, where deg(∗) is the algebraic degree of ∗. Let M(z) = ∏ m−1 j=0 mj(z) be the product of m mutuallyprime polynomials. Then m residues can be extracted from A(z), A(z) ⇔ (A(z) mod m0(z), A(z) mod m1(z),..., A(z) mod mm−1(z)). (1) The conversion from left to right in (1) is the evaluation of the residues of A(z) with respect to the residue system described by the factors of M(z). If deg(M) ≥ N, and on condition that the mj(z) are mutually prime, this conversion is invertible, and then the conversion from right to left in (1) is the reconstruction of A(z) from its residues by the Chinese remainder theorem (CRT). 1 In this paper we are particularly interested in moduli, M(z), which split completely into linear factors, i.e. such that deg(M) = m, in which case the residues of A(z) are complex numbers. Moreover, assuming m ≥ N, and that mj(z) = z − ej, the residues can be computed by the action of an m × m Vandermonde matrix such that
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.
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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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
1 PAPER Complementary Sequence Pairs of Types II and III ∗
"... SUMMARY Bipolar complementary sequence pairs of Types II and III are defined, enumerated for n ≤ 28, and classified. TypeII pairs are shown to exist only at lengths 2m, and necessary conditions for TypeIII pairs lead to a nonexistence conjecture regarding their length. ..."
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SUMMARY Bipolar complementary sequence pairs of Types II and III are defined, enumerated for n ≤ 28, and classified. TypeII pairs are shown to exist only at lengths 2m, and necessary conditions for TypeIII pairs lead to a nonexistence conjecture regarding their length.