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179
On approximately symmetric informationally complete positive operatorvalued measures and related systems of quantum states
 J. MATH. PHYS
, 2008
"... We address the problem of constructing positive operatorvalued measures (POVMs) in finite dimension n consisting of n 2 operators of rank one which have an inner product close to uniform. This is motivated by the related question of constructing symmetric informationally complete POVMs (SICPOVMs) ..."
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We address the problem of constructing positive operatorvalued measures (POVMs) in finite dimension n consisting of n 2 operators of rank one which have an inner product close to uniform. This is motivated by the related question of constructing symmetric informationally complete POVMs (SICPOVMs) for which the inner products are perfectly uniform. However, SICPOVMs are notoriously hard to construct and despite some success of constructing them numerically, there is no analytic construction known. We present two constructions of approximate versions of SICPOVMs, where a small deviation from uniformity of the inner products is allowed. The first construction is based on selecting vectors from a maximal collection of mutually unbiased bases and works whenever the dimension of the system is a prime power. The second construction is based on perturbing the matrix elements of a subset of mutually unbiased bases. Moreover, we construct vector systems in C n which are almost orthogonal and which might turn out to be useful for quantum computation. Our constructions are based on results of analytic number theory.
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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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.
Planar functions over fields of characteristic two
 J. Algebraic Combin
"... Abstract. Classical planar functions are functions from a finite field to itself and give rise to finite projective planes. They exist however only for fields of odd characteristic. We study their natural counterparts in characteristic two, which we also call planar functions. They again give rise t ..."
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Abstract. Classical planar functions are functions from a finite field to itself and give rise to finite projective planes. They exist however only for fields of odd characteristic. We study their natural counterparts in characteristic two, which we also call planar functions. They again give rise to finite projective planes, as recently shown by the second author. We give a characterisation of planar functions in characteristic two in terms of codes over Z4. We then specialise to planar monomial functions f(x) = cxt and present constructions and partial results towards their classification. In particular, we show that t = 1 is the only odd exponent for which f(x) = cxt is planar (for some nonzero c) over infinitely many fields. The proof techniques involve methods from algebraic geometry. 1.
New Sequences Design from Weil Representation with Low TwoDimensional Correlation in Both Time and Phase Shifts
"... For a given prime p, a new construction of families of the complex valued sequences of period p with efficient implementation is given by applying both multiplicative characters and additive characters of finite field Fp. Such a signal set consists of p 2 (p − 2) timeshift distinct sequences, the m ..."
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For a given prime p, a new construction of families of the complex valued sequences of period p with efficient implementation is given by applying both multiplicative characters and additive characters of finite field Fp. Such a signal set consists of p 2 (p − 2) timeshift distinct sequences, the magnitude of the twodimensional autocorrelation function (i.e., the ambiguity function) in both time and phase of each sequence is upper bounded by 2 √ p at any shift not equal to (0, 0), and the magnitude of the ambiguity function of any pair of phaseshift distinct sequences is upper bounded by 4 √ p. Furthermore, the magnitude of their Fourier transform spectrum is less than or equal to 2. A proof is given through finding a simple elementary construction for the sequences constructed from the Weil representation by Gurevich, Hadani and Sochen. An open problem for directly establishing these assertions without involving the Weil representation is addressed. Index Terms. Sequence, autocorrelation, cross correlation, ambiguity function, Fourier transform, and Weil representation. 1
Double Circulant Codes over Z4 and Even Unimodular Lattices
, 1997
"... With the help of some new results about weight enumerators of selfdual codes over Z4 we investigate a class of double circulant codes over Z4, one of which leads to an extremal even unimodular 40dimensional lattice. It is conjectured that there should be “Nine more constructions of the Leech latt ..."
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With the help of some new results about weight enumerators of selfdual codes over Z4 we investigate a class of double circulant codes over Z4, one of which leads to an extremal even unimodular 40dimensional lattice. It is conjectured that there should be “Nine more constructions of the Leech lattice”.
TwoLevel Nonregular Designs From Quaternary Linear Codes
, 2006
"... Abstract: A quaternary linear code is a linear space over the ring of integers modulo 4. Recent research in coding theory shows that many famous nonlinear codes such as the Nordstrom and Robinson (1967) code and its generalizations can be simply constructed from quaternary linear codes. This paper e ..."
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Abstract: A quaternary linear code is a linear space over the ring of integers modulo 4. Recent research in coding theory shows that many famous nonlinear codes such as the Nordstrom and Robinson (1967) code and its generalizations can be simply constructed from quaternary linear codes. This paper explores the use of quaternary codes to construct twolevel nonregular designs. A general construction of nonregular designs is described and some theoretic results are obtained. Many nonregular designs constructed by this method have better statistical properties than regular designs of the same size in terms of resolution and aberration. A systematic construction procedure is proposed and a collection of nonregular designs with 16, 32, 64, 128, 256 runs and up to 64 factors is presented. Key words and phrases: Fractional factorial design, generalized minimum aberration, generalized resolution, MacWilliams identity, quaternary code.
Cycle index, weight enumerator, and Tutte polynomial
 Electronic J. Combinatorics
, 2002
"... With every linear code is associated a permutation group whose cycle index is the weight enumerator of the code (up to a trivial normalisation). ..."
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With every linear code is associated a permutation group whose cycle index is the weight enumerator of the code (up to a trivial normalisation).
On generalized Hamming weights for Galois ring linear codes,
 Designs, Codes and Cryptography,
, 1998
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On the integrality of nth roots of generating functions
 J. COMBINATORIAL
, 2006
"... Motivated by the discovery that the eighth root of the theta series of the E8 lattice and the 24th root of the theta series of the Leech lattice both have integer coefficients, we investigate the question of when an arbitrary element f ∈ R (where R = 1 + xZ[[x]]) can be written as f = g n for g ∈ R, ..."
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Motivated by the discovery that the eighth root of the theta series of the E8 lattice and the 24th root of the theta series of the Leech lattice both have integer coefficients, we investigate the question of when an arbitrary element f ∈ R (where R = 1 + xZ[[x]]) can be written as f = g n for g ∈ R, n ≥ 2. Let Pn: = {g n  g ∈ R} and let µn: = n ∏ pn p. We show among other things that (i) for f ∈ R, f ∈ Pn ⇔ f (mod µn) ∈ Pn, and (ii) if f ∈ Pn, there is a unique g ∈ Pn with coefficients mod µn/n such that f ≡ gn (mod µn). In particular, if f ≡ 1 (mod µn) then f ∈ Pn. The latter assertion implies that the theta series of any extremal even unimodular lattice in Rn (e.g. E8 in R8) is in Pn if n is of the form 2i3j5k (i ≥ 3). There do not seem to be any exact analogues for codes, although we show that the weight enumerator of the rth order ReedMuller code of length 2m is in P2r (and similarly that the theta series of the BarnesWall lattice BW2m is in P2m). We give a number of other results and conjectures, and establish a conjecture of Paul D. Hanna that there is a unique element f ∈ Pn (n ≥ 2) with coefficients restricted to the set {1, 2,..., n}.