Post on 30-Jun-2020
Parallelisms of PG(3,4) with
automorphisms of order 3
Svetlana Topalova, Stela Zhelezova
Institute of Mathematics and Informatics, BAS, Bulgaria
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Parallelisms of PG(3,4) with automorphisms of order 3
Parallelisms – relations and applications
Definitions and notations
History
PG(3,4) and related 2-designs
Construction
Results
Johnson, Combinatorics of Spreads and Parallelisms, CRC Press (2010)
constant dimension error correcting codes that contain lifted MRD codes: Etzion,
Silberstein, Codes and designs related to lifted MRD codes, 2011;
anonymous (2, q + 1)-threshold schemes: Stinson, Combinatorial designs:
constructions and analysis, 2004;
wireless key predistribution schemes: Ruj, Seberry, Roy, Key predistribution
schemes using block designs in wireless sensor networks, 2009.
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Parallelisms of PG(3,4) with automorphisms of order 3
Parallelisms – relations and applications
Lunardon, On regular parallelisms in PG(3,q), 1984;
Walker, Spreads covered by derivable partial spreads, 1985;
Jha and Johnson, On regular r-packings, 1986 :
PG (2r − 1, q) and a translation plane of order q2r.
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Parallelisms of PG(3,4) with automorphisms of order 3
Regular parallelisms – relations application
regular parallelism in
PG (3,q)
spread in PG(7,q)
translation plane of order q4
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Parallelisms of PG(3,4) with automorphisms of order 3
Definitions and notations
Spread in PG(d,q) - a partition of the point set by lines.
Parallelism in PG(d,q) – a partition of the set of lines by spreads.
Automorphism group of the parallelism – maps each spread of the
parallelism to a spread of the same parallelism.
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Parallelisms of PG(3,4) with automorphisms of order 3
Regulus of PG(3,q) – a set R={l1, …, lq+1} of mutually skew lines, any
line l li = pi,
l lj = pj, l ls ≠ Ø , ls R
l lk = pk,
Regular spread S={S1, …, Sq2+1} of PG(3,q) :
R(li, lj, lk) S.
Regular parallelism – all its spreads are regular.
Definitions and notations
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2-design:
V – finite set of v points
B – finite collection of b blocks: k-element subsets of V
D = (V, B ) – 2-(v,k,λ) design if any 2-subset of V is in λ blocks of B.
Parallel class – a partition of the point set by blocks.
Resolution – a partition of the collection of blocks by parallel classes.
Parallelisms of PG(3,4) with automorphisms of order 3
Definitions and notations
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Parallelisms of PG(3,4) with automorphisms of order 3
General constructions of parallelisms:
in PG(n,2), Zaicev, Zinoviev, Semakov, 1973; Baker, 1976.
in PG(2n-1,q), Beutelspacher, 1974.
a pair of orthogonal parallelisms – Fuji-Hara in PG(3,q), 1986.
two infinite families of regular cyclic parallelisms, PG(3,q), q ≡ 2
(mod 3), Penttila and Williams, 1998.
History
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Parallelisms of PG(3,4) with automorphisms of order 3
Parallelisms in PG(3,q):
PG(3,2) – all (2) are classified, regular;
PG(3,3) – all (73 343) are classified, Betten, 2016;
PG(3,4) – with autom. of orders 5 or 7, Topalova, Zhelezova, 2013/2015;
PG(3,5) – cyclic parallelisms, Prince, 1998;
– regular parallelisms with autom. of order 3, Top., Zhel., 2016.
History
Parallelisms of PG(3,4) with automorphisms of order 3
Regular parallelisms in PG(3,q):
PG(3,q) – two infinite families of regular cyclic parallelisms,
q ≡ 2 (mod 3), Penttila and Williams, 1998.
PG(3,2) – all are regular.
PG(3,5) – cyclic regular parallelisms, Prince, 1998
– regular parallelisms with autom. of order 3, Topalova, Zhelezova, 2016
(not from the Penttila and Williams families)
PG(3,3) – at most 11regular S in a parallelism, Betten, 2016.
PG(3,4) – at most 11regular S in a parallelism (order 5),
Topalova, Zhelezova, 2015.
History
Bamberg, 2012;
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Parallelisms of PG(3,4) with automorphisms of order 3
The incidence of the points and t-dimensional subspaces of PG(d,q)
defines a 2-design (D).
points of D
blocks of D
resolutions of D
points of PG(3,4)
lines of PG(3,4)
parallelisms of PG(3,4)
2-(85,5,1) design
PG(3,4) and related 2-designs
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PG(3,4) points, lines.
G – group of automorphisms of PG(3,4):
|G| = 213.34.52.7.17
Gi – subgroup of order i.
Parallelisms of PG(3,4) with automorphisms of order 3
851
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q
qv
d
35711 22 qqq
PG(3,4) and related 2-designs
t-dimentional subspaces 1 ( lines ) 2 (hyperplanes)
2-(v,k,) design 2-(85,5,1)
b=357, r=21
2-(85,21,5)
b=85, r=21
Parallelisms of PG(3,4)
21 spreads with 17 lines
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Parallelisms of PG(3,4) with automorphisms of order 3
Sylow subgroup of order 81 (G81) - 3 conjugacy classes of its subgroups of
order 3 – GAP – http://www.gap-system.org
G3 – the subgroup of order 3 which yields parallelisms.
Construction
357 lines
27 fixed lines 110 orbits of
length 3
50 orbits with
nondisjoint lines
60 orbits with
disjoint lines
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Parallelisms of PG(3,4) with automorphisms of order 3
Construction of spreads:
• backtrack search on lines;
• n+1st line - contains the first point, which is in none of the n spread lines;
• fixed spread – add the whole line orbit;
• non fixed spread – lines are from different orbits of one and the same length;
• lexicographically ordered;
• orbit leader – a fixed spread or the first in lexicographic order spread from an
orbit under G3;
Construction
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Parallelisms of PG(3,4) with automorphisms of order 3
The types of orbit leaders under G3 :
• a spread of 2 fixed lines and 5 line orbits with disjoint lines (F2) – only one;
• a spread of 5 fixed lines and 4 line orbits with disjoint lines (F5) – 141 for each
fixed line containing the first point;
• a spread with an orbit of length three (O) – 29 624 disjoint to the fixed lines and
the F2 spread.
A parallelism invariant under G3
Construction
F2 F5 F5 F5 F5 F5 O O O O O
fixed part – 4959 nonisomorphic fixed parts
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Parallelisms of PG(3,4) with automorphisms of order 3
Isomorphic solutions rejection
N (G3) – normalizer of G3 in G
| N (G3)| = 43 200
The rejection place
}|{)( 31
33 GggGGgGN
Construction
fixed part – 4959 nonisomorphic fixed parts
F2 F5 F5 F5 F5 F5 O O O O O
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Parallelisms of PG(3,4) with automorphisms of order 3
R e s u l t s
|GP| 3 6 12 15 24 30 48 60 96 960
All 8 115 559 4 488 52 40 14 38 12 8 2 4
The order of the full automorphism group
of the parallelisms of PG(3,4) with automorphisms of order 3
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Parallelisms of PG(3,4) with automorphisms of order 3
R e s u l t s
R S A 3 6 12 15 24 30 48 60 96 960 All
1* 0 5*15 259 661 434 16 12 260 123
1* 5* 15 13 052 250 20 10 13 332
1* 15 5* 3 886 116 12 4 014
1* 5*15 0 866 124 52 4 6 4 12 8 2 4 1 082
1*6 1*6 4*3 286 4 290
1*12 1*3 4* 10 10
other types 7 837 798 3 560 8 7 841 366
All 8 115 559 4 488 52 40 14 38 12 8 2 4 8 120 217
The type of the spreads
of the parallelisms of PG(3,4) with automorphisms of order 3