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Spectral Orbits and PeaktoAverage Power Ratio of Boolean Functions with respect to the {I, H, N}^n Transform
 SETA’04, SEQUENCES AND THEIR APPLICATIONS, SEOUL, ACCEPTED FOR PROCEEDINGS OF SETA04, LECTURE NOTES IN COMPUTER SCIENCE
, 2005
"... We enumerate the inequivalent selfdual additive codes over GF(4) of blocklength n, thereby extending the sequence A090899 in The OnLine Encyclopedia of Integer Sequences from n = 9 to n = 12. These codes have a wellknown interpretation as quantum codes. They can also be represented by graphs, wh ..."
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Cited by 18 (14 self)
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We enumerate the inequivalent selfdual additive codes over GF(4) of blocklength n, thereby extending the sequence A090899 in The OnLine Encyclopedia of Integer Sequences from n = 9 to n = 12. These codes have a wellknown interpretation as quantum codes. They can also be represented by graphs, where a simple graph operation generates the orbits of equivalent codes. We highlight the regularity and structure of some graphs that correspond to codes with high distance. The codes can also be interpreted as quadratic Boolean functions, where inequivalence takes on a spectral meaning. In this context we define PARIHN, peaktoaverage power ratio with respect to the {I, H, N} n transform set. We prove that PARIHN of a Boolean function is equivalent to the the size of the maximum independent set over the associated orbit of graphs. Finally we propose a construction technique to generate Boolean functions with low PARIHN and algebraic degree higher than 2.
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.
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 8 (2 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.
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 7 (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
Spectral interpretations of the interlace polynomial
, 2005
"... We relate the interlace polynomials of a graph to the spectra of a quadratic boolean function with respect to a strategic subset of local unitary transforms. By so doing we establish links between graph theory, cryptography, coding theory, and quantum entanglement. We establish the form of the inter ..."
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Cited by 5 (1 self)
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We relate the interlace polynomials of a graph to the spectra of a quadratic boolean function with respect to a strategic subset of local unitary transforms. By so doing we establish links between graph theory, cryptography, coding theory, and quantum entanglement. We establish the form of the interlace polynomial for certain functions, provide a new interlace polynomial, QHN, and propose a generalisation of the interlace polynomial to hypergraphs. We also prove some conjectures from [13] and equate certain spectral metrics with various evaluations of the interlace polynomial.
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.
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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Cited by 3 (0 self)
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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