A Family of Optimal Packings in Grassmannian Manifolds
P.W. Shor
and N.J.A. Sloane
DOI: 10.1023/A:1008608404829
Abstract
A remarkable coincidence has led to the discovery of a family of packings of m 2 + m - 2\text m/2 m^2 + m - 2{\text{ }}m/2 -dimensional subspaces of m-dimensional space, whenever m is a power of 2. These packings meet the 
orthoplex bound 
and are therefore optimal.
orthoplex bound and are therefore optimal.Pages: 157–163
Keywords: Grassmannian manifold; packing; separating subspace; Barnes-wall lattice; quantum coding theory; Clifford group
Full Text: PDF
References
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2. E.S. Barnes and N.J.A. Sloane, “New lattice packings of spheres,” Canad. J. Math. 35 (1983), 117-130. P1: PMR Journal of Algebraic Combinatorics KL540-03-Shor February 5, 1998 10:46 A FAMILY OF OPTIMAL PACKINGS 163
3. E.S. Barnes and G.E. Wall, “Some extreme forms defined in terms of Abelian groups,” J. Australian Math. Soc. 1 (1959), 47-63.
4. B. Bolt, T.G. Room, and G.E. Wall, “On Clifford collineation, transform and similarity groups I,” J. Australian Math. Soc. 2 (1961), 60-79.
5. B. Bolt, T.G. Room, and G.E. Wall, “On Clifford collineation, transform and similarity groups II,” J. Australian Math. Soc. 2 (1961), 80-96.
6. W. Bosma and J. Cannon, Handbook of Magma Functions, Sydney, 1995.
7. W. Bosma, J.J. Cannon, and G. Mathews, “Programming with algebraic structures: Design of the Magma language,” in Proceedings of the 1994 International Symposium on Symbolic and Algebraic Computation, M. Giesbrecht (Ed.), Association for Computing Machinery, Oxford, 1994, pp. 52-57.
8. W. Bosma, J. Cannon, and C. Playoust, “The Magma algebra system I: The user language,” J. Symb. Comp. 1996, 24 (1997), 235-265.
9. A.R. Calderbank, P.J. Cameron, W.M. Kantor, and J.J. Seidel, “Z4-Kerdock codes, orthogonal spreads, and extremal Euclidean line-sets,” Proc. London Math. Soc., 75 (1997), 436-480.
10. A.R. Calderbank and P.W. Shor, “Good quantum error-correcting codes exist,” Phys. Rev. A, 54 (1996), 1098-1105.
11. J.H. Conway, R.H. Hardin, and N.J.A. Sloane, “Packing lines, planes, etc.: Packings in Grassmannian space,” Experimental Math. 5 (1996), 139-159.
12. J.H. Conway and N.J.A. Sloane, Sphere Packings, Lattices and Groups, 2nd edition, Springer-Verlag, NY, 1993.
13. G.D. Forney, Jr., “Coset codes-Part II: Binary lattices and related codes,” IEEE Trans. Information Theory 34 (1988), 1152-1187.
14. G.E. Wall, “On Clifford collineation, transform and similarity groups IV,” Nagoya Math. J. 21 (1962), 199- 222.