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Constructions of Mutually Unbiased Bases
 in Proc. 7th Int. Conf. on finite fields and applications, Lecture
"... Abstract. Two orthonormal bases B and B ′ of a ddimensional complex innerproduct space are called mutually unbiased if and only if 〈bb ′ 〉  2 = 1/d holds for all b ∈ B and b ′ ∈ B ′. The size of any set containing pairwise mutually unbiased bases of C d cannot exceed d + 1. If d is a power of ..."
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Cited by 22 (1 self)
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Abstract. Two orthonormal bases B and B ′ of a ddimensional complex innerproduct space are called mutually unbiased if and only if 〈bb ′ 〉  2 = 1/d holds for all b ∈ B and b ′ ∈ B ′. The size of any set containing pairwise mutually unbiased bases of C d cannot exceed d + 1. If d is a power of a prime, then extremal sets containing d+1 mutually unbiased bases are known to exist. We give a simplified proof of this fact based on the estimation of exponential sums. We discuss conjectures and open problems concerning the maximal number of mutually unbiased bases for arbitrary dimensions.
New construction of mutually unbiased bases in square dimensions
"... We show that k = w + 2 mutually unbiased bases can be constructed in any square dimension d = s 2 provided that there are w mutually orthogonal Latin squares of order s. The construction combines the designtheoretic objects (k, s)nets (which can be constructed from w mutually orthogonal Latin squa ..."
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Cited by 14 (2 self)
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We show that k = w + 2 mutually unbiased bases can be constructed in any square dimension d = s 2 provided that there are w mutually orthogonal Latin squares of order s. The construction combines the designtheoretic objects (k, s)nets (which can be constructed from w mutually orthogonal Latin squares of order s and vice versa) and generalized Hadamard matrices of size s. Using known lower bounds on the asymptotic growth of the number of mutually orthogonal Latin squares (based on number theoretic sieving techniques), we obtain that the number of mutually unbiased bases in dimensions d = s 2 is greater than s 1/14.8 for all s but finitely many exceptions. Furthermore, our construction gives more mutually orthogonal bases in many nonprimepower dimensions than the construction that reduces the problem to prime power dimensions. 1
Mutually Orthogonal Latin Squares: A Brief Survey of Constructions
, 1999
"... In the two centuries since Euler first asked about mutually orthogonal latin squares, substantial progress has been made. The biggest breakthroughs came in 1960 with the celebrated theorems of Bose, Shrikhande, and Parker, and in 1974 in the research of Wilson. Current efforts have concentrated on r ..."
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Cited by 8 (1 self)
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In the two centuries since Euler first asked about mutually orthogonal latin squares, substantial progress has been made. The biggest breakthroughs came in 1960 with the celebrated theorems of Bose, Shrikhande, and Parker, and in 1974 in the research of Wilson. Current efforts have concentrated on refining these approaches, and finding new applications of the substantial theory opened. This paper provides a detailed list of constructions for MOLS, concentrating on the uses of pairwise balanced designs and transversal designs in recursive constructions as pioneered in the papers of Bose, Shrikhande, and Parker. In addition, several new lower bounds for MOLS are given and an uptodate table of lower bounds for MOLS is provided. 1 An Historical Introduction In 1779, Euler began a study of a simple mathematical puzzle, the 36 Officers Problem. Thirtysix officers drawn from six different ranks and six different regiments (one of each rank from each regiment) are to be arranged in a squar...
Sieve Methods
"... Sieve methods have had a long and fruitful history. The sieve of Eratosthenes (around 3rd century B.C.) was a device to generate prime numbers. Later Legendre used it in his studies of the prime number counting function π(x). Sieve methods bloomed and became a topic of intense investigation after th ..."
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Sieve methods have had a long and fruitful history. The sieve of Eratosthenes (around 3rd century B.C.) was a device to generate prime numbers. Later Legendre used it in his studies of the prime number counting function π(x). Sieve methods bloomed and became a topic of intense investigation after the pioneering work of Viggo Brun (see [Bru16],[Bru19], [Bru22]). Using his formulation of the sieve Brun proved, that the sum
More Thwarts in Transversal Designs
, 1995
"... Using thwarts, new transversal designs are determined for orders 201, 336, 360, 365, 393, 429, 501, 749, 845, 1080, 1120, 1324, 1400, 1632, 1760, 1824, 1904, and for numerous larger orders. Incomplete transversal designs with block size eight, and PBDs having three consecutive block sizes, are al ..."
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Using thwarts, new transversal designs are determined for orders 201, 336, 360, 365, 393, 429, 501, 749, 845, 1080, 1120, 1324, 1400, 1632, 1760, 1824, 1904, and for numerous larger orders. Incomplete transversal designs with block size eight, and PBDs having three consecutive block sizes, are also constructed.