Geometry & Topology Volume 20, issue 3 (2016)

Download this article
	For screen
	For printing

Recent Issues

The Journal

Submission Guidelines

Submission Page

Policies for Authors

Ethics Statement

ISSN 1364-0380 (online)

ISSN 1465-3060 (print)

Author Index

To Appear

Other MSP Journals

Optimal simplices and codes in projective spaces

Henry Cohn, Abhinav Kumar and Gregory Minton

Geometry & Topology 20 (2016) 1289–1357

DOI: 10.2140/gt.2016.20.1289

Bibliography

1	C An, X Chen, I H Sloan, R S Womersley, Well conditioned spherical designs for integration and interpolation on the two-sphere, SIAM J. Numer. Anal. 48 (2010) 2135 MR2763659
2	E Anderson, Z Bai, C Bischof, L S Blackford, J Demmel, J Dongarra, J Du Croz, A Greenbaum, S Hammarling, A McKenney, D Sorenson, LAPACK Users’ Guide, SIAM (1999)
3	D M Appleby, I Bengtsson, S Brierley, Å Ericsson, M Grassl, J Å Larsson, Systems of imprimitivity for the Clifford group, Quantum Inf. Comput. 14 (2014) 339 MR3186299
4	D M Appleby, C A Fuchs, H Zhu, Group theoretic, Lie algebraic and Jordan algebraic formulations of the SIC existence problem, Quantum Inf. Comput. 15 (2015) 61 MR3362195
5	C Bachoc, Linear programming bounds for codes in Grassmannian spaces, IEEE Trans. Inform. Theory 52 (2006) 2111 MR2234468
6	C Bachoc, Y Ben-Haim, S Litsyn, Bounds for codes in products of spaces, Grassmann, and Stiefel manifolds, IEEE Trans. Inform. Theory 54 (2008) 1024 MR2445048
7	C Bachoc, R Coulangeon, G Nebe, Designs in Grassmannian spaces and lattices, J. Algebraic Combin. 16 (2002) 5 MR1941981
8	C Bachoc, G Nebe, F M de Oliveira Filho, F Vallentin, Lower bounds for measurable chromatic numbers, Geom. Funct. Anal. 19 (2009) 645 MR2563765
9	C Bachoc, F Vallentin, New upper bounds for kissing numbers from semidefinite programming, J. Amer. Math. Soc. 21 (2008) 909 MR2393433
10	J C Baez, The octonions, Bull. Amer. Math. Soc. 39 (2002) 145 MR1886087
11	E Bannai, S G Hoggar, On tight t–designs in compact symmetric spaces of rank one, Proc. Japan Acad. Ser. A Math. Sci. 61 (1985) 78 MR796472
12	E Bannai, S G Hoggar, Tight t–designs and squarefree integers, European J. Combin. 10 (1989) 113 MR988506
13	V Bargmann, Note on Wigner’s theorem on symmetry operations, J. Mathematical Phys. 5 (1964) 862 MR0164620
14	M Berger, A panoramic view of Riemannian geometry, Springer (2003) MR2002701
15	A V Bondarenko, On a spherical code in the space of spherical harmonics, Ukrainian Math. J. 62 (2010) 993 MR2888653
16	A Bondarenko, D Radchenko, M Viazovska, Optimal asymptotic bounds for spherical designs, Ann. of Math. (2) 178 (2013) 443 MR3071504
17	M J Bowick, L Giomi, Two-dimensional matter: order, curvature and defects, Adv. Phys. 58 (2009) 449
18	U Brehm, The shape invariant of triangles and trigonometry in two-point homogeneous spaces, Geom. Dedicata 33 (1990) 59 MR1042625
19	U Brehm, B Et-Taoui, Congruence criteria for finite subsets of complex projective and complex hyperbolic spaces, Manuscripta Math. 96 (1998) 81 MR1624356
20	U Brehm, B Et-Taoui, Congruence criteria for finite subsets of quaternionic elliptic and quaternionic hyperbolic spaces, Geom. Dedicata 84 (2001) 261 MR1825360
21	U Brehm, W Kühnel, 15–vertex triangulations of an 8–manifold, Math. Ann. 294 (1992) 167 MR1180457
22	A R Calderbank, R H Hardin, E M Rains, P W Shor, N J A Sloane, A group-theoretic framework for the construction of packings in Grassmannian spaces, J. Algebraic Combin. 9 (1999) 129 MR1679247
23	X Chen, Numerical verification methods for spherical t–designs, Japan J. Indust. Appl. Math. 26 (2009) 317 MR2589478
24	X Chen, A Frommer, B Lang, Computational existence proofs for spherical t–designs, Numer. Math. 117 (2011) 289 MR2754852
25	X Chen, R S Womersley, Existence of solutions to systems of underdetermined equations and spherical designs, SIAM J. Numer. Anal. 44 (2006) 2326 MR2272596
26	E A Coddington, N Levinson, Theory of ordinary differential equations, McGraw–Hill (1955) MR0069338
27	A M Cohen, Exceptional presentations of three generalized hexagons of order 2, J. Combin. Theory Ser. A 35 (1983) 79 MR704257
28	H Cohn, Order and disorder in energy minimization, from: "Proceedings of the International Congress of Mathematicians, IV" (editors R Bhatia, A Pal, G Rangarajan, V Srinivas, M Vanninathan), Hindustan Book Agency (2010) 2416 MR2827978
29	H Cohn, A Kumar, Universally optimal distribution of points on spheres, J. Amer. Math. Soc. 20 (2007) 99 MR2257398
30	H Cohn, Y Zhao, Energy-minimizing error-correcting codes, IEEE Trans. Inform. Theory 60 (2014) 7442 MR3285724
31	J H Conway, R H Hardin, N J A Sloane, Packing lines, planes, etc : packings in Grassmannian spaces, Experiment. Math. 5 (1996) 139 MR1418961
32	H S M Coxeter, Regular complex polytopes, Cambridge Univ. Press (1974) MR0370328
33	J Creignou, Constructions of Grassmannian simplices, arXiv:cs/0703036
34	P Delsarte, J M Goethals, J J Seidel, Spherical codes and designs, Geom. Dedicata 6 (1977) 363 MR0485471
35	N D Elkies, B H Gross, The exceptional cone and the Leech lattice, Internat. Math. Res. Notices (1996) 665 MR1411589
36	M Fickus, D G Mixon, J C Tremain, Steiner equiangular tight frames, Linear Algebra Appl. 436 (2012) 1014 MR2890902
37	L Fousse, G Hanrot, V Lefèvre, P Pélissier, P Zimmermann, MPFR: a multiple-precision binary floating-point library with correct rounding, ACM Trans. Math. Software 33 (2007) MR2326955
38	É Fouvry, E Kowalski, P Michel, Counting sheaves using spherical codes, Math. Res. Lett. 20 (2013) 305 MR3151649
39	H Hadwiger, Über ausgezeichnete Vektorsterne und reguläre Polytope, Comment. Math. Helv. 13 (1940) 90 MR0003718
40	T Hangan, G Masala, A geometrical interpretation of the shape invariant for geodesic triangles in complex projective spaces, Geom. Dedicata 49 (1994) 129 MR1266268
41	R H Hardin, N J A Sloane, McLaren’s improved snub cube and other new spherical designs in three dimensions, Discrete Comput. Geom. 15 (1996) 429 MR1384885
42	J M Hendrickx, A Olshevsky, Matrix p–norms are NP-hard to approximate if p≠1,2,∞, SIAM J. Matrix Anal. Appl. 31 (2010) 2802 MR2740634
43	S G Hoggar, t–designs in projective spaces, European J. Combin. 3 (1982) 233 MR679208
44	S G Hoggar, Parameters of t–designs in 𝔽P^d−1, European J. Combin. 5 (1984) 29 MR746042
45	S G Hoggar, Tight t–designs and octonions, Mitt. Math. Sem. Giessen 165 (1984) 1 MR745865
46	W Hurewicz, H Wallman, Dimension theory, 4, Princeton Univ. Press (1941) MR0006493
47	G A Kabatjanskiĭ, V I Levenšteĭn, Bounds for packings on the sphere and in space, Problemy Peredači Informacii 14 (1978) 3 MR0514023
48	W M Kantor, Quaternionic line-sets and quaternionic Kerdock codes, Linear Algebra Appl. 226/228 (1995) 749 MR1344596
49	L V Kantorovich, On Newton’s method for functional equations, Doklady Akad. Nauk SSSR 59 (1948) 1237 MR0024700
50	M Khatirinejad Fard, Regular structures of lines in complex spaces, PhD thesis, Simon Fraser University (2008) MR2736698
51	H König, Cubature formulas on spheres, from: "Advances in multivariate approximation" (editors W Haußmann, K Jetter, M Reimer), Math. Res. 107, Wiley-VCH (1999) 201 MR1797231
52	S G Krantz, H R Parks, The implicit function theorem: history, theory, and applications, Birkhäuser (2003) MR1894435
53	R Krawczyk, Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken, Computing (Arch. Elektron. Rechnen) 4 (1969) 187 MR0255046
54	P W H Lemmens, J J Seidel, Equiangular lines, J. Algebra 24 (1973) 494 MR0307969
55	V I Levenshteĭn, Bounds on the maximal cardinality of a code with bounded modulus of the inner product, Dokl. Akad. Nauk SSSR 263 (1982) 1303 MR653223
56	V I Levenshteĭn, Designs as maximum codes in polynomial metric spaces, Acta Appl. Math. 29 (1992) 1 MR1192833
57	J H van Lint, J J Seidel, Equilateral point sets in elliptic geometry, Nederl. Akad. Wetensch. Proc. Ser. A 69 (1966) 335 MR0200799
58	Y I Lyubich, On tight projective designs, Des. Codes Cryptogr. 51 (2009) 21 MR2480685
59	D G Mixon, C J Quinn, N Kiyavash, M Fickus, Fingerprinting with equiangular tight frames, IEEE Trans. Inf. Theory 59 (2013) 1855 MR3030758
60	N Mukunda, R Simon, Quantum kinematic approach to the geometric phase. I. General formalism, Ann. Physics 228 (1993) 205 MR1252254
61	J W Neuberger, The continuous Newton’s method, inverse functions, and Nash–Moser, Amer. Math. Monthly 114 (2007) 432 MR2309983
62	A Neumaier, Combinatorial configurations in terms of distances, Memorandum 81-09 (Wiskunde en Informatica) (1980)
63	J M Ortega, The Newton–Kantorovich theorem, Amer. Math. Monthly 75 (1968) 658 MR0231218
64	, PARI/GP version 2.6.0 (2013)
65	J M Renes, Equiangular spherical codes in quantum cryptography, Quantum Inf. Comput. 5 (2005) 81 MR2123901
66	J M Renes, Equiangular tight frames from Paley tournaments, Linear Algebra Appl. 426 (2007) 497 MR2350673
67	J M Renes, R Blume-Kohout, A J Scott, C M Caves, Symmetric informationally complete quantum measurements, J. Math. Phys. 45 (2004) 2171 MR2059685
68	N Revol, F Rouillier, Motivations for an arbitrary precision interval arithmetic and the MPFI library, Reliab. Comput. 11 (2005) 275 MR2158338
69	A J Scott, M Grassl, Symmetric informationally complete positive-operator-valued measures: a new computer study, J. Math. Phys. 51 (2010) MR2662471
70	M A Sustik, J A Tropp, I S Dhillon, R W Heath Jr., On the existence of equiangular tight frames, Linear Algebra Appl. 426 (2007) 619 MR2350682
71	R R Thomas, Lectures in geometric combinatorics, 33, Amer. Math. Soc. (2006) MR2237292
72	S Waldron, On the construction of equiangular frames from graphs, Linear Algebra Appl. 431 (2009) 2228 MR2567829
73	L R Welch, Lower bounds on the maximum cross correlation of signals, IEEE Trans. Inf. Theory 20 (1974) 397
74	W K Wootters, B D Fields, Optimal state-determination by mutually unbiased measurements, Ann. Physics 191 (1989) 363 MR1003014
75	P Xia, S Zhou, G B Giannakis, Achieving the Welch bound with difference sets, IEEE Trans. Inf. Theory 51 (2005) 1900 MR2235693
76	G Zauner, Quantum designs: foundations of a noncommutative design theory, Int. J. Quantum Inf. 9 (2011) 445 MR2931102