There are fairly large families of unitarily inequivalent complete sets of N + 1 mutually unbiased bases (MUBs) known in |$\mathbb{C}$|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 well known concerning known families of complete sets of MUBs and their associated planes.
REFERENCES
1.
Albert
, A. A.
, “Generalized twisted fields
,” Pacific J. Math.
11
, 1
–8
(1961
).2.
Alltop
, W. O.
, “Complex sequences with low periodic correlations
,” IEEE Trans. Inf. Theory
26
, 350
–354
(1980
).3.
Bader
, L.
, Kantor
, W. M.
, and Lunardon
, G.
, “Symplectic spreads from twisted fields
,” Boll. Unione Mat. Ital.
8-A
, 383
–389
(1994
).4.
Ball
, S.
, Bamberg
, J.
, Lavrauw
, M.
, and Penttila
, T.
, “Symplectic spreads
,” Designs, Codes, Cryptogr.
32
, 9
–14
(2004
).5.
Bandyopadhyay
, S.
, Boykin
, P. O.
, Roychowdhury
, V.
, and Vatan
, F.
, “A new proof for the existence of mutually unbiased bases
,” Algorithmica
34
, 512
–528
(2002
).6.
Bengtsson
, I.
, “MUBs, polytopes, and finite geometries
,” in Foundations of Probability and Physics
, edited by A.
Khrennikov
, AIP Conference Proceedings
Vol. 750
(AIP
, Melville, NY
, 2005
), Vol. 3, pp. 63
–69
.7.
Bierbrauer
, J.
, “Commutative semifields from projection mappings
,” Des. Codes Cryptogr.
61
, 187
–196
(2011
).8.
Bierbrauer
, J.
and Kantor
, W. M.
, “A projection construction for semifields
” (submitted).9.
Boykin
, P. O.
, Sitharam
, M.
, Tarifi
, M.
, and Wocjan
, P.
, “Real mutually unbiased bases
,” e-print arXiv:quant-ph/0502024v2.10.
Boykin
, P. O.
, Sitharam
, M.
, Tiep
, P. H.
, and Wocjan
, P.
, “Mutually unbiased bases and orthogonal decompositions of Lie algebras
,” Quantum Inf. Comput.
7
, 371
–382
(2007
).11.
Budaghyan
, L.
and Carlet
, C.
, “Classes of quadratic APN trinomials and hexanomials and related structures
,” IEEE Trans. Inf. Theory
54
, 2354
–2357
(2008
).12.
Budaghyan
, L.
and Helleseth
, T.
, “New commutative semifields defined by new PN multinomials
,” Cryptogr. Commun.
3
, 1
–16
(2011
).13.
Calderbank
, A. R.
, “Reed Muller codes and symplectic geometry
,” in Recent Trends in Coding Theory and its Applications
, edited by W.-C.
Li
, AMS/IP Studies in Advanced Mathematics
Vol. 41
(AMS
, Providence
, 2007
), pp. 123
–147
.14.
Calderbank
, A. R.
, Cameron
, P. J.
, Kantor
, W. M.
, and Seidel
, J. J.
, “|${\mathbb z}$|4-Kerdock codes, orthogonal spreads, and extremal Euclidean line-sets
,” Proc. LMS
75
, 436
–480
(1997
).15.
Cohen
, S. D.
and Ganley
, M. J.
, “Commutative semifields, two–dimensional over their middle nuclei
,” J. Algebra
75
, 373
–385
(1982
).16.
Coulter
, R. S.
and Matthews
, R. W.
, “Planar functions and planes of Lenz-Barlotti class II
,” Designs, Codes, Cryptogr.
10
, 167
–184
(1997
).17.
Delsarte
, P.
, Goethals
, J. M.
, and Seidel
, J. J.
, “Bounds for systems of lines, and Jacobi polynomials
,” Philips Res. Rep.
30
, 91
–105
(1975
).18.
19.
Dillon
, J. F.
, “On Pall partitions for quadratic forms
” (unpublished).20.
Ding
, C.
and Yuan
, J.
, “A family of skew Paley-Hadamard difference sets
,” J. Comb. Theory Ser. A
113
, 1526
–1535
(2006
).21.
Dye
, R. H.
, “Partitions and their stabilizers for line complexes and quadrics
,” Ann. Mat. Pura. Appl.
114
, 173
–194
(1977
).22.
Ganley
, M. J.
, “Central weak nucleus semifields
,” Eur. J. Comb.
2
, 339
–347
(1981
).23.
Godsil
, C.
and Roy
, A.
, “Equiangular lines, mutually unbiased bases, and spin models
,” Eur. J. Comb.
30
, 246
–262
(2009
).24.
Howe
, R.
, “Nice error bases, mutually unbiased bases, induced representations, the Heisenberg group and finite geometries
,” Indag. Math.
16
, 553
–583
(2005
).25.
Ivanović
, I. D.
, “Geometrical description of quantal state determination
,” J. Phys. A
14
, 3241
–3245
(1981
).26.
Kantor
, W. M.
, “Spreads, translation planes and Kerdock sets. I, II
,” SIAM J. Algebraic Discrete Methods
3
, 151
–165
, 308–318 (1982
).27.
Kantor
, W. M.
, “Ovoids and translation planes
,” Can. J. Math.
34
, 1195
–1207
(1982
).28.
Kantor
, W. M.
, “Projective planes of order q whose collineation groups have order q2
,” J. Algebr. Comb.
3
, 405
–425
(1994
).29.
Kantor
, W. M.
, “Codes, quadratic forms and finite geometries
,” in Different Aspects of Coding Theory
, edited by A. R.
Calderbank
, Proceedings of Symposia in Applied Mathematics
Vol. 50
(American Mathematical Society
, Providence
, 1995
), pp. 153
–177
.30.
Kantor
, W. M.
, “Quaternionic line-sets and quaternionic Kerdock codes
,” Linear Algebr. Appl.
226–228
, 749
–779
(1995
).31.
Kantor
, W. M.
, “Note on Lie algebras, finite groups and finite geometries
,” in Groups, Difference Sets, and the Monster
, edited by K. T.
Arasu
et al (de Gruyter
, Berlin
, 1996
), pp. 73
–81
.32.
Kantor
, W. M.
, “Commutative semifields and symplectic spreads
,” J. Algebra
270
, 96
–114
(2003
).33.
Kantor
, W. M.
and Williams
, M. E.
, “New flag-transitive affine planes of even order
,” J. Comb. Theory Ser. A
74
, 1
–13
(1996
).34.
Kantor
, W. M.
and Williams
, M. E.
, “Symplectic semifield planes and |${\mathbb z}$|4-linear codes
,” Trans. AMS
356
, 895
–938
(2004
).35.
Kantor
, W. M.
and Williams
, M. E.
, “Nearly flag-transitive affine planes
,” Adv. Geom.
10
, 161
–183
(2010
).36.
König
, H.
, “Isometric imbeddings of Euclidean spaces into finite-dimensional lp-spaces
,” in Banach Center Publications
(PWN-Polish Scientific
, Warsaw
, 1995
), Vol. 34
, pp. 79
–87
.37.
Lunardon
, G.
, Marino
, G.
, Polverino
, O.
, and Trombetti
, R.
, “Symplectic semifield spreads of PG(5, q) and the Veronese surface
,” Ric. Mat.
60
, 125
–142
(2011
).38.
Penttila
, T.
and Williams
, B.
, “Ovoids of parabolic spaces
,” Geom. Dedic.
82
, 1
–19
(2000
).39.
Roy
, A.
and Scott
, A. J.
, “Weighted complex projective 2-designs from bases: optimal state determination by orthogonal measurements
,” J. Math. Phys.
48
, 072110
(2007
).40.
Seidel
, J. J.
, “Harmonics and combinatorics
,” in Special Functions: Group Theoretical Aspects and Applications
, edited by R. A.
Askey
et al (Reidel
, Dordrecht
, 1984
), pp. 287
–303
.41.
Scharlau
, R.
and Tiep
, P. H.
, “Symplectic groups, symplectic spreads, codes, and unimodular lattices
,” J. Algebra
194
, 113
–156
(1997
).42.
Strohmer
, T.
, Heath
, R. W.
Jr., and Paulraj
, A. J.
, “On the design of optimal spreading sequences for CDMA systems
,” in The Thirty-Sixth Asilomar Conference on Signals, Systems & Computers
, edited by M. B.
Matthews
(IEEE
, 2002
), pp. 1434
–1438
.43.
Thas
, J. A.
and Payne
, S. E.
, “Spreads and ovoids in finite generalized quadrangles
,” Geom. Dedic.
52
, 227
–253
(1994
).44.
Tits
, J.
“Ovoïdes et groupes de Suzuki
,” Arch. Math.
13
, 187
–198
(1962
).45.
Wootters
, W. K.
, “A Wigner-function formulation of finite-state quantum mechanics
,” Ann. Phys.
176
, 1
–21
(1987
).46.
Wootters
, W. K.
and Fields
, B. D.
“Optimal state-determination by mutually unbiased measurements
,” Ann. Phys.
191
, 363
–381
(1989
).47.
Zha
, Z.
and Wang
, X.
“New families of perfect nonlinear polynomial functions
,” J. Algebra
322
, 3912
–3918
(2009
).48.
We do not use bases since automorphisms do not preserve bases (e.g., Z(b) in (2.1) does not preserve the standard basis).
49.
We avoid the term frame used in Ref. 14 so as not to conflict with other uses for that word.
50.
An affine plane of order N is a combinatorial object consisting of a set of N2points, together with N2 + N point-sets of size N called lines, such that any two distinct points are on a unique line. Then the lines fall into N + 1 “parallel classes” of size N, each of which partitions the points.
51.
See the delectable observation at the end of Ref. 6.
52.
We are identifying isomorphic vector spaces.
53.
This means that f(x + a) − f(x) = b has a unique solution x for any a ≠ 0 and b in K.
54.
Curiously, there is also a group of N2 automorphisms of π(f) that does not act on
${\mathcal F}^f$
. (This is the group of all (x, y) ↦ (x + c, y + d), c, d ∈ K, having orbits of size N and N2 on the set of all lines.)© 2012 American Institute of Physics.
2012
American Institute of Physics
You do not currently have access to this content.