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A GroupTheoretic Framework for the Construction of Packings in Grassmannian Spaces
, 2002
"... By using totally isotropic subspaces in an orthogonal space Ω + (2i,2), several infinite families of packings of 2 kdimensional subspaces of real 2 idimensional space are constructed, some of which are shown to be optimal packings. A certain Clifford group underlies the construction and links this ..."
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By using totally isotropic subspaces in an orthogonal space Ω + (2i,2), several infinite families of packings of 2 kdimensional subspaces of real 2 idimensional space are constructed, some of which are shown to be optimal packings. A certain Clifford group underlies the construction and links this problem with BarnesWall lattices, Kerdock sets and quantumerrorcorrecting codes.
MUBS INEQUIVALENCE AND AFFINE PLANES
, 2011
"... There are fairly large families of unitarily inequivalent complete sets of N+1 mutually unbiased bases (MUBs) in C N for various prime powers N. The number of such sets is not bounded above by any polynomial as a function of N. While it is standard that there is a superficial similarity between com ..."
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There are fairly large families of unitarily inequivalent complete sets of N+1 mutually unbiased bases (MUBs) in C N for various prime powers N. The number of such sets is not bounded above by any polynomial as a function of N. While it is standard that there is a superficial similarity between complete sets of MUBs and finite affine planes, there is an intimate relationship between these large families and affine planes. This note briefly summarizes “old ” results that do not appear to be wellknown concerning known families of complete sets of MUBs and their associated planes.
Orthogonal spreads and translation planes
"... There have hccn a nmnhcr of striking new fesults concerning translation planes of characteristic 2, ohtained using orthogonal and sYlnplcctic spreads. The iInpdus for this came from coding theory. This paper surveys the gCOlndric advances, while providing a hint of their coding theoretic connections ..."
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Cited by 3 (0 self)
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There have hccn a nmnhcr of striking new fesults concerning translation planes of characteristic 2, ohtained using orthogonal and sYlnplcctic spreads. The iInpdus for this came from coding theory. This paper surveys the gCOlndric advances, while providing a hint of their coding theoretic connections.
A GroupTheoretic Framework for the Construction of Packings in Grassmannian Spaces
, 1997
"... Abstract. By using totally isotropic subspaces in an orthogonal space � +(2i, 2), several infinite families of packings of 2kdimensional subspaces of real 2idimensional space are constructed, some of which are shown to be optimal packings. A certain Clifford group underlies the construction and li ..."
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Abstract. By using totally isotropic subspaces in an orthogonal space � +(2i, 2), several infinite families of packings of 2kdimensional subspaces of real 2idimensional space are constructed, some of which are shown to be optimal packings. A certain Clifford group underlies the construction and links this problem with BarnesWall lattices, Kerdock sets and quantumerrorcorrecting codes. Keywords: Grassmannian packings, quantum computing, orthogonal geometry, Clifford group
OPTIMAL SIMPLICES AND CODES IN PROJECTIVE SPACES
"... Abstract. We find many tight codes in compact spaces, i.e., optimal codes whose optimality follows from linear programming bounds. In particular, we show the existence (and abundance) of several hitherto unknown families of simplices in quaternionic projective spaces and the octonionic projective pl ..."
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Abstract. We find many tight codes in compact spaces, i.e., optimal codes whose optimality follows from linear programming bounds. In particular, we show the existence (and abundance) of several hitherto unknown families of simplices in quaternionic projective spaces and the octonionic projective plane. The most noteworthy cases are 15point simplices in HP2 and 27point simplices in OP2, both of which are the largest simplices and the smallest 2designs possible in their respective spaces. These codes are all universally optimal, by a theorem of Cohn and Kumar. We also show the existence of several positivedimensional families of simplices in the Grassmannians of subspaces of Rn with n ≤ 8; close numerical approximations to these families had been found by Conway, Hardin, and Sloane, but no proof of existence was known. Our existence proofs are computerassisted, and the main tool is a variant of the NewtonKantorovich theorem. This effective implicit function theorem shows, in favorable conditions, that every approximate solution to a set of polynomial equations has a nearby exact solution. Finally, we also exhibit a few explicit codes, including a configuration of 39 points in OP2 which form a maximal system of mutually unbiased bases. This is the last tight code in OP2 whose existence had been previously conjectured but not resolved. Contents