Des. Codes Cryptogr. DOI 10.1007/s10623-013-9887-3

On point-transitive and transitive deficiency one parallelisms of PG(3, 4) Svetlana Topalova · Stela Zhelezova

Received: 16 December 2012 / Revised: 19 September 2013 / Accepted: 3 October 2013 © Springer Science+Business Media New York 2013

Abstract A parallelism in PG(n, q) is point-transitive if it has an automorphism group which is transitive on the points. If the automorphism group fixes one spread and is transitive on the remaining spreads, the parallelism corresponds to a transitive deficiency one parallelism. It is known that there are three types of spreads in PG(3, 4)—regular, subregular and aregular. A parallelism is regular if all its spreads are regular. In PG(3, 4) no point-transitive parallelisms, no regular ones, and no transitive deficiency one parallelisms have been known. Both pointtransitive parallelisms and transitive deficiency one parallelisms must have automorphisms of order 5. We construct all 32,048 nonisomorphic parallelisms with automorphisms of order 5 and classify them by the orders of their automorphism groups and by the types of their spreads. There are 31,832 parallelisms with an automorphism group fixing exactly one spread. Only for four of them the automorphism group is transitive on the remaining spreads. Among the parallelisms we construct there are no regular ones. There are 4,124 parallelisms with automorphisms of order 5 without fixed points, but none of them is point-transitive. Keywords

Spread · Parallelism · Transitivity · Automorphisms

Mathematics Subject Classification

05B05 · 51E20 · 51E23

1 Introduction The relation to translation planes [9] has been one of the main reasons for the consideration of t-spreads and t-parallelisms in PG(n, q). It is also important that sets of mutually orthogonal parallelisms of PG(3, q) might lead to a projective plane of order q(q + 1) [4,14]. There

Communicated by L. Teirlinck. S. Topalova · S. Zhelezova (B) Institute of Mathematics and Informatics, BAS, P.O. Box 323, 5000 Veliko Tarnovo, Bulgaria e-mail: [email protected] S. Topalova e-mail: [email protected]

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are applications in other areas too. The relation of parallelisms to resolutions of Steiner systems leads to a cryptographic usage for anonymous (2, q + 1)-threshold schemes [28], while in Coding Theory t-parallelisms are used in constructions of constant dimension errorcorrecting codes that contain lifted MRD codes [10,26]. 1.1 Definitions For the basic concepts and notations concerning spreads and parallelisms of projective spaces, refer, for instance, to [9], [15] or [30]. A t-spread in PG(n, q) is a set of distinct t-dimensional subspaces which partition the point set. A partial t-parallelism is a set of mutually disjoint t-spreads. A t-parallelism is a partition of the set of t-dimensional subspaces by t-spreads. A deficiency one t-parallelism is a partial t-parallelism with one t-spread less than the t-parallelism. Each deficiency one t-parallelism can be uniquely extended to a t-parallelism. Usually 1-spreads (1-parallelisms) are called line spreads (line parallelisms) or just spreads (parallelisms). There can be line spreads and parallelisms if n is odd. Two parallelisms are isomorphic if there exists an automorphism of the projective space which maps each spread of the first parallelism to a spread of the second one. A subgroup of the automorphism group of the projective space, which maps each spread of the (partial) parallelism to a spread of the same (partial) parallelism is called an automorphism group of the (partial) parallelism. A (partial) parallelism is called transitive if it has an automorphism group, which is transitive on the spreads. A transitive deficiency one parallelism corresponds to a parallelism with an automorphism group which fixes one spread and is transitive on the remaining spreads. The incidence of the points and t-dimensional subspaces of PG(n, q) defines a 2-design (see for instance [30, 2.35–2.36]). There is a one-to-one correspondence between the parallelisms of PG(3, 4) and the resolutions of the 2-(85,5,1) point-line design. 1.2 Parallelisms 1.2.1 General results A construction of parallelisms in PG(n, 2) is presented by Zaitsev et al. [34] and independently by Baker [1], and in PG(2n − 1, q) by Beutelspacher [2]. Constructions in PG(3, q) are known due to Denniston [5], Johnson [12,13], Penttila and Williams [20]. 1.2.2 Computer-aided classifications Several computer aided classifications of t-parallelisms are available too. Prince classified parallelisms of PG(3, 3) with automorphisms of order 5 [21], and parallelisms of PG(3, 5) with automorphisms of order 31 [22]. Stinson and Vanstone [29] classified parallelisms of PG(5, 2) with a full automorphism group of order 155, Sarmiento [24] with a point-transitive cyclic group of order 63, and Zhelezova [35] with a cyclic group of order 31. Topalova and Zhelezova [31] classified 2-parallelisms of PG(5, 2) with automorphisms of order 31 and Sarmiento [25] with a cyclic group of order 63. 1.2.3 Transitivity Examples of transitive 1-parallelisms of PG(3, q) are presented in [5,7,20,22] and of PG(5, 2) in [29,35]. The only known examples of transitive t-parallelisms for t > 1 are the

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On point-transitive and transitive deficiency of PG(3, 4)

2-parallelisms in PG(5, 2) [31]. Transitivity and double transitivity is considered by Johnson [15] and by Johnson and Montinaro [16,17] who show that only two doubly-transitive parallelisms exist, and these are all the parallelisms in PG(3, 2). They determine the structure of the automorphism group of a transitive parallelism of PG(n, q) [17] and point out that transitive t-parallelisms in PG(n, q) can only exist for t = 1, or for t = 2 and (n, q) = (5, 2) or (n, q) = (5, 3). An infinite class of transitive deficiency one parallelisms of PG(3, q) is provided by Johnson [13] for q = pr if p is odd and further a group-theoretic characterization of the constructed parallelisms is presented by Johnson and Pomareda [18]. Properties of the automorphism groups and the spreads of transitive deficiency one parallelisms of PG(3, q) are derived by Biliotti et al. [3], and Diaz et al. [8], who show that the deficiency spread must be Desarguesian, and the automorphism group should contain a normal subgroup of order q 2 (see also [15, Chap. 38]). Part of the present results were announced in a conference paper [32], where we erroneously claim that there are no transitive deficiency one parallelisms in PG(3, 4). This was because of an error in our computation of the orders of the automorphism groups. 1.2.4 PG(3, 4) Denniston proved that there are no cyclic parallelisms in PG(3, 4) [6]. It was shown in [33] that transitive parallelisms in PG(3, 4) cannot occur. No examples of transitive deficiency one parallelisms of PG(3, 4) were known before the present work. 1.2.5 The present result A deficiency one parallelism in PG(3, 4) has 20 spreads. That is why the order of a transitive automorphism group must be divisible by 20, and therefore there must be automorphisms of order 5. There are 85 points in PG(3, 4). That is why a point-transitive parallelism must have an automorphism of order 5 too. We construct all parallelisms of PG(3, 4) with automorphisms of order 5. We classify them by the order of the full automorphism group. None of them is point-transitive. Four of the parallelisms we construct yield the first examples of transitive deficiency one parallelisms in PG(3, 4). 1.3 Regularity A regulus of PG(2t +1, q) is a set R of q +1 mutually skew lines such that any line intersecting three elements of R intersects all elements of R. Such a line is called transversal. All the transversals of a regulus form its opposite regulus. A spread S of PG(2t + 1, q) is regular if for every three distinct elements of S, the unique regulus determined by them is a subset of S. A spread is called aregular [9] if it contains no regulus and subregular or semi-regular if it can be obtained from a regular spread by successive replacements of some reguli by their opposites. We will call a parallelism uniform if all its spreads are of one and the same type. A uniform parallelism is regular (subregular, aregular) if its spreads are regular (subregular, aregular). Pentilla and Williams [20] constructed two regular cyclic parallelisms of PG(3, q) for any q ≡ 2 (mod 3). All known examples of regular parallelisms are among them. There are three nonisomorphic spreads in PG(3, 4) [27]. One of them is regular (Desarguesian), one subregular, and one aregular.

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We classify all the constructed parallelisms by the orders of their automorphism groups and by the types of their spreads. There are no regular ones among them. 1.4 Research method We find a generating set of the automorphism group of PG(3, 4) as generating set of the automorphism group of the related 2-(85,21,5) point-hyperplane design. Next we choose the necessary subgroups of order 5 and compute their normalizers. Our C++ programmes constructing the parallelisms are based on the exhaustive backtrack search techniques (see for instance [19, Chap. 4]). To filter away isomorphic parallelisms, we use the normalizer of the predefined automorphism group in the automorphism group of the projective space.

2 Construction 2.1 The automorphism group of PG(3, 4) There are 85 points and 357 lines in PG(3, 4). Denote by G the full automorphism group of PG(3, 4). Hence, G ∼ = PL(4, 4), whose order is 213 · 34 · 52 · 7 · 17. A spread has 17 lines which partition the point set and a parallelism has 21 spreads. To construct PG(3, 4) we use G F(4) with elements 0, 1, ω and ω2 , where ω is a root of the polynomial x 2 + x + 1 = 0. The points of PG(3, 4) are then all 4-dimensional vectors (v1 , v2 , v3 , v4 ) over G F(4) such that vi = 1 if i is the maximum index for which vi  = 0. We sort these 85 vectors in ascending lexicographic order and then assign them numbers such that (1, 0, 0, 0) is number 1, and (ω2 , ω2 , ω2 , 1) number 85. Each invertible matrix (ai, j )4×4 over G F(4) defines an automorphism of this projective  space by the map vi = ai, j v j . The following 9 automorphism matrices will be referred j

to in the text below. ⎞ ⎛ 2 ⎛ ⎛ ⎞ 1 1 ω2 ω ω 0 0 ω 0 0 ω2 ⎜1 ω 0 0⎟ ⎜0 ω 1 1 ⎟ ⎜ ω2 ω 1 ⎟ ⎜ ⎜ ⎟ A0,17 = ⎜ ⎝ 0 0 1 1 ⎠ A0,2 = ⎝ 1 ω 1 ω ⎠ A5,2 = ⎝ 0 1 ω 0 0 ω2 0 0 ω2 1 ω2 1 ω2 ω2 ⎞ ⎞ ⎞ ⎛ ⎛ ⎛ 1 0 ω2 0 1 0 ω2 ω2 1 ω2 0 1 ⎜ 0 1 ω2 ω ⎟ ⎜ 0 1 ω ω2 ⎟ ⎜0 1 1 0 ⎟ ⎟ ⎟ ⎟ ⎜ ⎜ B1 = ⎜ ⎝ 0 0 ω 0 ⎠ B2 = ⎝ 0 0 ω 0 ⎠ C1 = ⎝ 0 0 ω ω2 ⎠ 0 0 0 ω 0 0 0 ω 0 0 1 0 ⎞ ⎞ ⎞ ⎛ ⎛ ⎛ 2 1 ω ω 0 1 ω2 ω ω2 1 ω2 ω2 ω 2 2 2 2 ⎜0 1 ω ω ⎟ ⎜0 1 ω 1 ⎟ ⎜0 1 0 ω ⎟ ⎟ ⎟ ⎟ ⎜ ⎜ C2 = ⎜ ⎝ 0 0 ω 0 ⎠ C 3 = ⎝ 1 0 0 0 ⎠ C 4 = ⎝ 0 0 ω2 1 ⎠ 0 ω 0 0 0 0 ω2 ω 0 0 ω 0

⎞ 1 0 ⎟ ⎟ 1 ⎠ ω2

2.2 Subgroups of order 5 A Sylow 5-subgroup of G has order 52 and all these subgroups are conjugate under G. We use GAP [11] to find an arbitrary Sylow 5-subgroup G 25 . It partitions the lines in 21 orbits, namely 13 orbits of length 25, 6 orbits of length 5, and 2 fixed lines. That is why the existence of parallelisms invariant under G 25 is impossible.

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On point-transitive and transitive deficiency of PG(3, 4)

G 25 has 6 subgroups of order 5 which are in three conjugacy classes. An arbitrary subgroup (of G) of order 5 is in one of these three conjugacy classes. So we take a subgroup of order 5 from each conjugacy class, namely the subgroup G 50,17 which fixes no points and 17 lines and has a generator defined by A0,17 , the subgroup G 50,2 with no fixed points, 2 fixed lines and generator A0,2 , and G 55,2 with 5 fixed points, 2 fixed lines and generator A5,2 . Below we consider one by one the results about G 50,17 , G 50,2 , and G 55,2 . In the next subsections we denote by G 5 any of these subgroups when we consider properties holding for all of them. 2.3 Construction of the parallelisms We sort the 357 lines (blocks of the 2-(85,5,1) design) in lexicographic order defined on the numbers of the points they contain and assign to each line a number according to this order. The spreads are constructed by backtrack search. If there are already n elements in the spread, we choose the n + 1-st one among the lines which have not been added to the parallelism yet and which contain the first point that is in none of the n spread elements. Adding a line to the parallelism, we also add to it all lines of its orbit under G 5 . A spread which is fixed by G 5 contains the whole orbit of any of its lines. If a spread is not fixed, we choose for it lines with orbits of one and the same length. Each spread is lexicographically greater than the ones constructed before it. For each spread, which is not fixed by G 5 , we already know the other spreads of its orbit. We call the first one orbit leader (a fixed spread is also an orbit leader). To obtain a parallelism we need to construct only the orbit leaders. 2.4 Isomorphism test The rejection of isomorphic solutions is an important part of the computation. We construct only parallelisms which are invariant under G 5 . Therefore we have to check if there is some permutation ϕ ∈ G such that it maps a parallelism P with G 5 to another parallelism with G 5 . Then P is invariant both under G 5 and under ϕ −1 G 5 ϕ. The Sylow 5-subgroups of the stabilizer in G of P are of order 5, and that is why there is an automorphism ψ of P such that ϕ −1 G 5 ϕ = ψ G 5 ψ −1 . It follows that ϕψ is in the normalizer N (G 5 ) of G 5 in G, which is defined as N (G 5 ) = {g ∈ G | gG 5 g −1 = G 5 } . Since ϕψ P = ϕ P , to establish isomorphism of two of the constructed parallelisms it is enough to consider only automorphisms from N (G 5 ). For each parallelism P we obtain, we check if an automorphism of N (G 5 ) maps it to a parallelism with a lexicographically smaller orbit leader sequence, and drop it if so. 2.5 Automorphism groups If some element of N(G 5 ) maps P to itself, it is its automorphism. So by the isomorphism test we also obtain some of the automorphisms of this parallelism. If P has an automorphism ψ∈ / N(G 5 ), then it is invariant under ψ G 5 ψ −1 . That is why we make a list of the generators of all conjugate groups of G 5 and check if some of them map P to itself. In fact we establish that only four of all constructed parallelisms are invariant under conjugate groups and they yield the transitive deficiency one parallelisms.

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S. Topalova and S. Zhelezova Table 1 The order of the full automorphism group of the parallelisms

|GP |

G 50,17 type 1

G 50,17 type 2

G 50,2

G 55,2

All

5

156

216

31,830

3,622

27,836

10

8

28

40

76

15

4

24

12

40

22

30

52

28

2

38

60

6

2

8

960

2

2

4

3,732

27,924

32,048

20 30

All

8

176

216

2.6 Software We obtain G 25 , the conjugacy classes of its subgroups, and the normalizers of G 50,17 , G 50,2 , and G 55,2 by GAP [11]. The other computer results are obtained by our own C++ programs. Most of the results are checked by two different programmes (algorithms) developed independently by the two authors.

3 Classification results All the results are summarised in Table 1, where the different orders of the full automorphism groups of the obtained parallelisms are presented in column |GP |. In the next four columns we give the number of parallelisms with this order of the full automorphism group, which are obtained with G 50,17 , G 50,2 , and G 55,2 . All the constructed parallelisms are available online at http://www.moi.math.bas.bg/~svetlana. 3.1 Automorphisms of order 5 with no fixed points and 17 fixed lines The group G 50,17 fixes 17 lines. The remaining 340 lines form 68 orbits of length 5. Two types of parallelisms invariant under G 50,17 are possible: 1. Type 1: Parallelisms with – one fixed spread made of the 17 fixed lines and – four orbits of five spreads each. 2. Type 2: Parallelisms with – one fixed spread with 7 fixed lines and 2 orbits of 5 lines and – five fixed spreads with 2 fixed lines and 3 orbits of 5 lines and – three orbits of five spreads each. The normalizer N (G 50,17 ) is a group of order 81600. We obtain 176 parallelisms of type 1, and 216 of type 2. A classification by the types of their spreads is presented in Tables 2 and 3. In Tables 2, 3, 4 and 5 we mark by * (star) the fixed spreads. 3.2 Automorphism of order 5 with no fixed points and 2 fixed lines The group G 50,2 fixes 2 lines. The remaining 355 lines form 71 orbits of length 5. The parallelisms invariant under G 50,2 have:

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On point-transitive and transitive deficiency of PG(3, 4) Table 2 Types of spreads of the parallelisms with G 50,17 , type 1

Parallelisms with |GP |

Types of spreads

Regular Subregular Aregular 5 1*

Table 3 Types of spreads of the parallelisms with G 50,17 , type 2

All

10 15 30

20

4

1*

5

15

80

1*

10

10

40

1*

15

5

16

1*

20

4

4

176

4

16 80 40

4

4

24

16

16 |GP | = 5, All

Types of spreads Regular

Subregular

Aregular

216

5*

1*

15

40

5*

1* + 5

10

72

5*

1* + 10

5

56

5* + 5

1*

5* + 5

1* + 5

10

8

5

40

Table 4 Types of spreads of the parallelisms with G 50,2 Parallelisms with |GP |

Types of spreads Regular

Subregular

Aregular

5

10

15

All 20

30

60

960

3,732

1*

20

554

4

558

1* + 5

15

1,028

2

1,030

1* + 10

10

760

10

4

774

1* + 15

5

270

4

4

278

1* + 20 1*

22

6

28

20

228

12

8

248

12

8

404

1*

5

15

384

1*

10

10

238

1*

15

5

84

1*

20

5

1*

15

16

16

5

1* + 5

10

12

12

1* + 5

5

10

10

1* + 5

10

5

4

10

1* + 5

1* + 10

5

12

238 2

2

8

6

4

94 6

2

32

2

12

5

2

2

5

2

2

4

– one fixed spread containing 2 fixed lines and 3 orbits of 5 lines and – four orbits of five spreads each. The normalizer N (G 50,2 ) is a group of order 600. We construct 3,732 parallelisms (Table 1). The types of their spreads are presented in Table 4.

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S. Topalova and S. Zhelezova Table 5 Types of spreads of the parallelisms with G 55,2 Parallelisms with |GP |

Types of spreads Regular

Subregular

Aregular

1*

20

3,696

2

1* + 5

15

7,695

4

1* + 10

10

6,344

8

1* + 15

5

2,391 324

20

1,178

1* + 20 1*

5

10

1*

5

15

2,634

1*

10

10

2,014

1*

15

5

1*

20

5

1*

15

132

5

1* + 5

10

176

5

1* + 10

5

104

5

1* + 15

1* + 5

15

All 20

30

60

960

27,924 3,698

3

7,702

16

7

2,414

2

2

6,352 328 1,178 8

4

2

2,644

4

815

2,022

8

114

4

2

823 2

2

124 132 176

4

4

112

30

30

15

52

52

1* + 5

5

10

56

56

1* + 5

10

5

53

53

1* + 5

15

8

8

10

1*

10

10

10

10

1* + 5

1* + 10 1* + 10

5

5

6

6

10

2

2

5

2

2

3.3 Automorphism of order 5 with 5 fixed points and 2 fixed lines The group G 55,2 fixes 2 lines. The remaining 355 lines form 71 orbits of length 5. The parallelisms invariant under G 55,2 have: – one fixed spread containing 2 fixed lines and 3 orbits of 5 lines and – four orbits of five spreads each. The normalizer N (G 55,2 ) is a group of order 3600. The number of nonisomorphic parallelisms is 27,924 (Table 1). A classification by the spread types is presented in Table 5. 3.4 Regularity One can see from the slanted rows of Tables 2, 4 and 5 that there are 6226 uniform deficiency one parallelisms, namely 528 subregular, and 6226 aregular ones. Most interesting among them are those 1614 whose fixed spread is regular (bold rows of the tables), because according to [8] transitive deficiency one parallelisms can occur only among them. There are 356 uniform parallelisms (row 5 of Tables 4 and 5), all of type subregular.

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On point-transitive and transitive deficiency of PG(3, 4) Table 6 The lines of the spreads S1 and S2

S1

S2

(1, 6,7, 8, 9)

(1, 6,7, 8, 9)

(2, 38, 42, 46, 50)

(2, 39, 43, 47, 51)

(3, 71, 74, 81, 84)

(3, 72, 77, 78, 83)

(4, 56, 58, 63, 69)

(4, 54, 60, 65, 67)

(5, 25, 26, 32, 35)

(5, 25, 26, 32, 35)

(10, 28, 40, 68, 80)

(10, 24, 44, 64, 84)

(11, 24, 45, 62, 83)

(11, 37, 48, 59, 70)

(12, 33, 51, 54, 76)

(12, 31, 53, 56, 74)

(13, 34, 49, 59, 72)

(13, 27, 40, 66, 81)

(14, 23, 47, 67, 75)

(14, 30, 38, 58, 82)

(15, 30, 39, 60, 85)

(15, 23, 46, 69, 76)

(16, 27, 53, 64, 70)

(16, 28, 50, 63, 73)

(17, 36, 43, 57, 78)

(17, 34, 45, 55, 80)

(18, 37, 41, 65, 77)

(18, 29, 49, 57, 85)

(19, 29, 48, 55, 82)

(19, 36, 41, 62, 75)

(20, 22, 52, 61, 79)

(20, 22, 52, 61, 79)

(21, 31, 44, 66, 73)

(21, 33, 42, 68, 71)

Table 7 Transitive deficiency one parallelisms of PG(3, 4)

Spread

|GP |

Generators of GP

P1

S1

960

A0,2

B1

f C1

P2

S2

960

A0,2

B2

f C2

P3

S1

960

A5,2

B1

f C3

P4

S2

960

A5,2

B2

f C4

3.5 Transitivity None of the automorphism group orders of the obtained parallelisms is divisible by 85, so no point-transitive parallelisms exist in PG(3, 4). In this subsection we denote lines of the projective space by a 5-tuple of the numbers of their points. G 50,2 and G 55,2 fix the same two lines, namely (11, 34, 47, 60, 73) and (19, 27, 46, 57, 84). The deficiency spread S0 of all the four transitive deficiency one parallelisms is the same. It is regular and contains the two fixed lines and the orbits under G 50,2 (G 55,2 ) of lines (1, 2, 3, 4, 5), (6, 22, 38, 54, 70) and (7, 33, 48, 63, 78). The lines of two particular spreads S1 and S2 are presented in Table 6. We present the four transitive deficiency one parallelisms in Table 7, where one of the spreads and generators of the full automorphism group are given. The notation f Ci means that the Frobenius automorphism (mapping ω to ω2 and vice versa) is applied after the transformation by the matrix Ci . The structure of these four parallelisms and their stabilizers is very similar, but they are not isomorphic, and their automorphism groups are not conjugate. 3.6 On correctness and relations to other papers – Each of the constructed transitive deficiency one parallelisms has a full automorphism group with a normal subgroup of order q 2 (16) and its deficiency spread is regular and not

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S. Topalova and S. Zhelezova

– – – –

–

isomorphic to the remaining spreads. These facts comply with the theoretical considerations of Biliotti et al. [3] and Diaz et al. [8]. We obtain the order of the normalizers of the chosen subgroups of order 5 both by our software and by GAP [11]. We obtain by two different C++ programmes (written by the two authors) the number of non-isomorphic parallelisms with G 50,17 type 1, G 50,2 , and G 55,2 . As a partial test for correctness of the automorphism group orders we obtain the stabilizer of the fixed spread of each parallelism P and check which of its elements map P to itself. To test our software for determining the spread type we find the type of all 5096448 distinct spreads in PG(3, 4) and obtain the same number of regular, subregular and aregular spreads as in [23]. To check the result about point-transitive parallelisms we try to construct parallelisms in PG(3, 4) admitting automorphisms of order 17 and establish that there are no such ones.

3.7 Final Remarks The classification results show that besides the four groups of order 960 there are other relatively rich parallelism automorphism groups too. The full automorphism groups of orders 5 or 10 partition the spreads of these parallelisms in 5 orbits (of lengths 1,5,5,5,5), those of order 20 in 4 orbits (of lengths 1,5,5,10), and the ones of orders 15, 30 or 60 in 3 orbits (of lengths 1,5,15). We believe that the present computer classification results for the parallelisms of PG(3, 4) with automorphisms of order 5 will be helpful to researchers who might seek theoretically for new constructions of parallelisms (deficiency one parallelisms) in other projective spaces. Acknowledgments We would like to thank the anonymous referees for the very careful reading of the paper, the important remarks and the adequate suggestions on the presentation of the material. This work was partially supported by the Bulgarian National Science Fund under Contract No. I01-0003.

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22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32.

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