972

# 2-( v , k , λ) Designs of Small Order

Authored by: Rudolf Mathon , Alexander Rosa

# Handbook of Combinatorial Designs

Print publication date:  November  2006
Online publication date:  November  2006

Print ISBN: 9781584885061
eBook ISBN: 9781420010541

10.1201/9781420010541-4

#### Abstract

1.1

A balanced incomplete block design (BIBD) is a pair ( V , B ) where V is a v-set and B is a collection of b k-subsets of V (blocks) such that each element of V is contained in exactly r blocks and any 2-subset of V is contained in exactly λ blocks. The numbers v, b, r, k, and λ are parameters of the BIBD.

#### 1.1  Definition and Basics

1.1

A balanced incomplete block design (BIBD) is a pair $( V , B )$ where V is a v-set and $B$ is a collection of b k-subsets of V (blocks) such that each element of V is contained in exactly r blocks and any 2-subset of V is contained in exactly λ blocks. The numbers v, b, r, k, and λ are parameters of the BIBD.

1.2 Proposition Trivial necessary conditions for the existence of a BIBD(v, b, r, k, λ) are (1) vr = bk, and (2) r(k − 1) = λ(v − 1). Parameter sets that satisfy (1) and (2) are admissible.

1.3

A BIBD $( X , D )$ is a subdesign of a BIBD $( V , B )$ if XV and $D ⊆ B$ . The subdesign is proper if XV.

1.4 Proposition If a (v, k, λ) design has a proper (w, k, λ) subdesign, then $w ≤ v − 1 k − 1$ .

1.5 Remark The three parameters v, k, and λ determine the remaining two as $r = λ ( v − 1 ) k − 1$ and $b = v r k$ . Hence one often writes (v, k, λ) design to denote a BIBD(v, b, r, k, λ). The notation 2-(v, k, λ) design is also used, because BIBDs are t-designs with t = 2. See §II.4. When λ = 1, the notation S(2, k, v) is also employed in the literature, because such BIBDs are Steiner systems. See §II.5. The notations S λ(2, k, v) or (v, k, λ) BIBD for a (v, k, λ) design are also in common use.

1.6

A BIBD $( V , B )$ with parameters v, b, r, k, λ is

 complete or full if it is simple and contains $( v k )$ blocks. decomposable if $B$ can be partitioned into two nonempty collections $B 1$ and $B 2$ so that $( V , B i )$ is a (v, k, λ i ) design for i = 1, 2. derived (from a symmetric design) $( X , D )$ if for some $D ∈ D$ , the collection of blocks ${ D ′ ∩ D : D ′ ∈ D , D ≠ D ′ } = B$ . Hadamard if v = 4n − 1, k = 2n − 1, and λ = n − 1 for some integer n ≥ 2. See §V.1. m-multiple if $v , b m , r m , k , λ m$ are the parameters of a BIBD. nontrivial if 3 ≤ k < v. quasi-symmetric if every two distinct blocks intersect in either μ 1 or μ 2 elements; the block intersection graph is strongly regular. See §VI.48 and §VI.11. residual (of a symmetric design) $( X , D )$ if for some $D ∈ D$ , the collection of blocks ${ D ′ \ D : D ′ ∈ D , D ≠ D ′ } = B$ . resolvable (an RBIBD) if there exists a partition R of its set of blocks B into parallel classes, each of which in turn partitions the set V; R is a resolution. See §II.7 simple if it has no repeated blocks. symmetric if v = b, or equivalently k = r. See §II.6.

1.7

The incidence matrix of a BIBD $( V , B )$ with parameters v, b, r, k, λ is a v × b matrix A = (a ij ), in which a ij = 1 when the ith element of V occurs in the jth block of $B$ , and a ij = 0 otherwise.

1.8 Theorem If A is the incidence matrix of a (v, k, λ)-design, then AA T = (r − λ)I + λJ and $J A = k J ^$ , where I is a v × v identity matrix, J is a v × v all ones matrix, and $J ^$ is a v × b all ones matrix. Moreover, any matrix A satisfying these conditions also satisfies λ(v − 1) = r(k − 1) and bk = vr; when k < v, it is the incidence matrix of a (v, k, λ) design. (See §VII.7.3)

1.9 Theorem (Fisher’s inequality) If a BIBD(v, b, r, k, λ) exists with 2 ≤ k < v, then bv.

1.10 Proposition An additional trivial necessary condition for the existence of an RBIBD is (3) k|v. A nontrivial condition is that bv + r − 1, a result that extends the inequality in Theorem 1.9. See §II.7.3.

1.11

Two BIBDs (V 1, B 1), (V 2, B 2) are isomorphic if there exists a bijection α : V 1V 2 such that B 1 α = B 2. Isomorphism of resolutions of BIBDs is defined similarly. An automorphism is an isomorphism of a design with itself. The set of all automorphisms of a design forms a group, the (full) automorphism group. An automorphism group of the design is any subgroup of the full automorphism group.

1.12 Remark If $( V , B )$ is a BIBD(v, b, r, k, λ) with automorphism group G, the action of G partitions $B$ into classes (orbits). A set of orbit representatives is a set of starter blocks or base blocks. Together with the group G, a set of base blocks can be used to define a design.

1.13

A BIBD $( V , B )$ with parameters v, b, r, k, λ and automorphism group G is

 cyclic if G contains a cycle of length v. regular if G contains a subgroup G′ of order v that acts transitively on the elements. k-rotational if some automorphism has one fixed point and k cycles each of length (v − 1)/k. transitive if for any two elements x and y, there is an automorphism mapping x to y.

1.14

The complement of a design $( V , B )$ is $( V , B ¯ )$ where $B ¯ = { V \ B : B ∈ B }$ .

1.15 Proposition The complement of a design with parameters (v, b, r, k, λ) is a design with parameters (v, b, br, vk, b − 2r + λ).

1.16 Remarks

1. In view of Proposition 1.15, one usually considers designs with v ≥ 2k and obtains the rest by taking the complement.
2. Complement has also been used, when $( V , B )$ is a simple 2-(v, b, r, k, λ) design, to denote the $2 − ( v , ( v k ) − b , ( v − 1 k − 2 ) − r , k , ( v − 2 k − 2 ) − λ )$ design obtained by taking all k-subsets not in $B$ as blocks. The term supplement is used here for this second kind of complementation.

#### 1.2  Small Examples

1.17 Remark To conserve space, designs are displayed in a k × b array in which each column contains the elements (taken from the decimal digits and roman letters) forming a block. For a discussion of how to find the automorphism group of each of these designs, see Remarks VII.6.107.

1.18 Example The unique (6, 3, 2) design and the unique (7, 3, 1) design (see also Example II.6.4).

 0000011122 0001123 1123423433 1242534 2345554545 3654656

1.19 Table The four nonisomorphic (7, 3, 2) designs.

 1: 00000011112222 2: 00000011112222 11335533443344 11335533443344 22446655666655 22446656565656 3: 00000011112222 4: 00000011112223 11334533453344 11234523453344 22456646565656 23456664565656

1.20 Table The 10 nonisomorphic (7, 3, 3) designs.

 1: 000000000111111222222 2: 000000000111111222222 111333555333444333444 111333555333444333444 222444666555666666555 222444666556566566556 3: 000000000111111222222 4: 000000000111111222222 111333455333445333444 111333455333445333444 222445666456566566556 222445666466556556566 5: 000000000111111222222 6: 000000000111111222223 111333445333445333445 111233455233445333444 222456566456566456566 223445666546566566565 7: 000000000111111222223 8: 000000000111111222223 111233455233445333444 111233445233455333444 223445666645566566556 223455666644566566556 9: 000000000111111222223 10: 000000000111111222233 111233445233445333454 111223445223345334544 223456566546566456665 234356566465656456656

1.21 Table The four nonisomorphic (8, 4, 3) designs.

 1: 00000001111222 2: 00000001111222 11123342334334 11123342334334 22554666455455 22554666455455 34675777677766 34675777767676 3: 00000001111222 4: 00000001111224 11123342334334 11122332233335 22554566465455 24645454545466 34677677577676 35767767667577

1.22 Example The unique (9, 3, 1) design.

 000011122236 134534534547 268787676858

1.23 Table The 36 nonisomorphic (9, 3, 2) designs.

 1: 000000001111112222223344 2: 000000001111112222223344 113355773344663344556655 113355773344663344556655 224466885577888866777788 224466885578787866787878 3: 000000001111112222223344 4: 000000001111112222223344 113355673344673344556655 113355773344663344555656 224467885568787867687878 224466885758786768788877 5: 000000001111112222223344 6: 000000001111112222223344 113355773344663344555656 113355673344673344555656 224466885758786867787887 224467885657887868678787 7: 000000001111112222223344 8: 000000001111112222223344 113355673344673344555656 113355673344673344555656 224467885658787867687887 224467885658787867688778 9: 000000001111112222223344 10: 000000001111112222223344 113355673344663344555756 113355773344563344565656 224467885858776767886878 224466885768786857877887 11: 000000001111112222223344 12: 000000001111112222223344 113355673344573344565656 113355673344573344565656 224467885678687856788787 224467885678687856877887 13: 000000001111112222223344 14: 000000001111112222223344 113355673344573344565656 113355673344573344565656 224467885867686758788787 224467885867686758877887 15: 000000001111112222223344 16: 000000001111112222223344 113346673345573344555656 113345673345673344556655 224557884676887868677887 224567884586787867687878 17: 000000001111112222223344 18: 000000001111112222223344 113345673345673344555656 113345673345663344555756 224567884856786778687887 224567884857786778686887 19: 000000001111112222223344 20: 000000001111112222223344 113345673345663344555657 113345673345563344566755 224567884857787867686788 224567884876785678877868 21: 000000001111112222223344 22: 000000001111112222223344 113345673345563344576655 113345673345563344575656 224567884876785867687788 224567884786876857687887 23: 000000001111112222223344 24: 000000001111112222233333 113345673345563344565756 112445672445674455644556 224567884876786758876887 233567883567887868778687 25: 000000001111112222233333 26: 000000001111112222233334 112445672445674455644556 112445672345673455644565 233567883568787867878687 233567884567888768778678 27: 000000001111112222233334 28: 000000001111112222233334 112445672345673455644565 112445672345673455644565 233567884568787868778687 233567884568788767878678 29: 000000001111112222233334 30: 000000001111112222233334 112445672345673455644565 112445672345673455644565 233567884576886878778876 233567884576888678778678 31: 000000001111112222233334 32: 000000001111112222233334 112445672345673455644565 112445672345673455644565 233567884586787768868877 233567884586788768767788 33: 000000001111112222233344 34: 000000001111112222233344 112355672344673345645655 112355672344673345645556 234467885568787868776878 234467885658786778887867 35: 000000001111112222233344 36: 000000001111112222233344 112345672345673345645655 112345672345673345645655 234567885468787876886778 234567885468787886776878

1.24 Table The 11 nonisomorphic (9, 4, 3) designs.

 1: 000000001111122223 2: 000000001111122223 111233562334433444 111233562334433444 225544777556666555 225544777565656565 346678888787878786 346678888788787786 3: 000000001111122223 4: 000000001111122223 111233462334533444 111233462334533444 225545777456666555 225545777456666555 346768888788778687 346768888877878687 5: 000000001111122223 6: 000000001111122223 111233462334533444 111233462334533444 225545677457666555 225545677457666555 346778888688778687 346788788678878687 7: 000000001111122223 8: 000000001111122223 111233462334533444 111233462334533444 225545777465656565 225545676465757565 346768888788787687 346778888877868876 9: 000000001111122223 10: 000000001111122223 111233462334533444 111233452334533444 225545676465757565 225546677467656565 346788788778868786 346787888578888677 11: 000000001111122234 111223352233533446 246454674745656557 357867886887887768

1.25 Table The three nonisomorphic (10, 4, 2) designs.

 1: 000000111122233 2: 000000111122233 3: 000000111122233 112356234534544 112356234534544 112345234534545 244778767658656 244778767658656 246867867647667 356899899899787 356899998889797 357989979858998

1.26 Table The unique (11, 5, 2) design and the unique (13, 4, 1) design.

 00000111223 0000111223345 11234236354 1246257364789 24567457465 385a46b57689a 35889898677 9c7ba8cb9cabc 769aaaa99a8

1.27 Table The two nonisomorphic (13, 3, 1) designs.

 1: 00000011111222223334445556 2: 00000011111222223334445556 13579b3469a3467867868a7897 13579b3469a3467867868a7897 2468ac578bc95acbbacc9bbac9 2468ac578bc95abcbcac9babc9

1.28 Table The 80 nonisomorphic (15, 3, 1) designs.

1.29 Table Properties of the 80 STS(15)s ((15,3,1) BIBDs). |G| is the order of the automorphism group. CI is the chromatic index, the minimum number of colors with which the blocks can be colored so that no two intersecting blocks receive the same color; when CI = 7, the design is resolvable. PC is the number of parallel classes. Sub is the number of (7,3,1)-subdesigns, and Pa is the number of Pasch configurations (four triples on six points).

#

|G|

CI

PC

Sub

Pa

#

|G|

CI

PC

Sub

Pa

#

|G|

CI

PC

Sub

Pa

1

20160

7

56

15

105

2

192

8

24

7

73

3

96

9

8

3

57

4

8

9

8

3

49

5

32

8

16

3

49

6

24

8

12

3

37

7

288

7

32

3

33

8

4

9

4

1

37

9

2

9

2

1

31

10

2

9

6

1

31

11

2

9

6

1

23

12

3

9

1

1

32

13

8

9

4

1

33

14

12

9

0

1

37

15

4

8

8

1

25

16

168

9

0

1

49

17

24

8

12

1

25

18

4

9

4

1

25

19

12

7

16

1

17

20

3

9

1

1

20

21

3

9

1

1

20

22

3

8

4

1

17

23

1

9

1

0

18

24

1

9

0

0

19

25

1

9

1

0

20

26

1

9

0

0

23

27

1

9

3

0

14

28

1

9

2

0

15

29

3

9

0

0

19

30

2

9

3

0

14

31

4

9

5

0

18

32

1

9

2

0

13

33

1

9

1

0

12

34

1

9

1

0

12

35

3

9

0

0

13

36

4

9

1

0

10

37

12

9

5

0

6

38

1

9

4

0

9

39

1

9

1

0

12

40

1

9

0

0

13

41

1

9

1

0

12

42

2

9

5

0

8

43

6

9

3

0

10

44

2

9

1

0

8

45

1

9

2

0

9

46

1

9

2

0

7

47

1

9

1

0

10

48

1

9

1

0

8

49

1

9

2

0

7

50

1

8

7

0

6

51

1

9

2

0

9

52

1

9

0

0

9

53

1

9

1

0

10

54

1

9

2

0

11

55

1

9

2

0

9

56

1

9

1

0

8

57

1

8

4

0

5

58

1

9

3

0

8

59

3

9

0

0

13

60

1

9

6

0

7

61

21

7

7

1

14

62

3

9

0

0

7

63

3

8

6

0

7

64

3

9

3

0

10

65

1

9

2

0

7

66

1

9

3

0

6

67

1

8

4

0

5

68

1

9

1

0

6

69

1

8

4

0

5

70

1

9

2

0

9

71

1

9

2

0

5

72

1

9

4

0

5

73

4

9

9

0

6

74

4

9

3

0

8

75

3

8

6

0

7

76

5

9

1

0

10

77

3

9

1

0

2

78

4

9

9

0

6

79

36

8

17

0

6

80

60

9

11

0

0

1.30 Table The five nonisomorphic (15, 7, 3) designs.

 1: 000000011112222 2: 000000011112222 3: 000000011112222 111335533443344 111335533443344 111335533443344 222446655666655 222446655666655 222446655666655 37b797978787878 37b797978787878 37b797978787878 48c8a8a9a9aa9a9 48c8a8a9a9aa9a9 48c8a8a9aa99aa9 59dbddbbccbbccb 59dbddbbccbcbbc 59dbddbbccbcbbc 6aeceecdeededde 6aeceecdeeddeed 6aeceecdedeeded 4: 000000011112222 5: 000000011112222 111335533443344 111335533443344 222446655666655 222446656565656 37b797879787878 37b797977888877 48c8a9a8aa99aa9 48c8a8a9aa99aa9 59dbdbcdbcbcbbc 59dbddbbccbcbbc 6aecedeecdeeded 6aeceecdedeeded

1.31 Example The unique (16, 4, 1) design.

1.32 Table The three nonisomorphic (16, 6, 2) designs.

 1: 0000001111222334 2: 0000001111222334 3: 0000001111222334 1123452345345455 1123452345345455 1123452345345455 2667896789877666 2667896789877666 2667896789877666 37aabcdbaa998987 37aabcdbaa998987 37aabcdbaa998987 48bddeeccbabccba 48bddeeccbacbbca 48bddeeccbcabbac 59ceffffedfeddef 59ceffffedfdeedf 59ceffffeddfeefd

1.33 Table The six nonisomorphic (19, 9, 4) designs.

 1: 0000000001111122223 2: 0000000001111122223 1111233562334433444 1111233562334433444 2225544777556666555 2225544777556666555 3346678888787878786 3346678888787878786 499acab99a9b9aa9b99 499acab99a9ba99ab99 5aebdcdabddcbcbccaa 5aebdcdabcdcbcbdcaa 6bfegefcdefecdfedbd 6bfegefdcffeddeedbc 7cgfhghfehgghfgfege 7cgfhghefhgggehfegf 8dhiiiihgiihighifih 8dhiiiihgiihhiigfih 3: 0000000001111122223 4: 0000000001111122223 1111233562334433444 1111233562334433444 2225544777565656565 2225544777565656565 3346678888788787786 3346678888788787786 499acab99a99abba999 499acab99a9ab99ba99 5aebdcdabbcdcccdbaa 5aebdcdabdbccccdbaa 6bfegefdcgfeddeedcb 6bfegefdcegfddeecbd 7cgfhghfehhgefffefg 7cgfhghfehhgeffffge 8dhiiiighiiihgghiih 8dhiiiighiihgiighih 5: 0000000001111122223 6: 0000000001111122223 1111233462334533444 1111233452334533444 2225545777456666555 2225546677467656565 3346768888788778687 3346787888578888677 499aca9a9abb999ab99 499aaab99bc9a99ab99 5aebdcdbbcdccabdcaa 5aebcddbcddcbabccaa 6bfeegfdcffeddeedbc 6bfegefcdefecdfeddb 7cgfghhefhggeghffge 7cgfhgghehhgfggfefe 8dhiiiihgiihihighif 8dhiiihifiiighihgih

1.34 Table The 18 nonisomorphic (25, 4, 1) designs. Their respective automorphism group orders are: 504,63,9,9,9,150,21,6,3,3,3,3,3,3,3,3,1,1.

#### 1.3  Parameter Tables

1.35 Table Admissible parameter sets of nontrivial BIBDs with r ≤ 41 and kv/2. Earlier listings of BIBDs by Hall [1016], Takeuchi [1999], and Kageyama [1241] and papers by Hanani [1042] and Wilson [2144] are frequently referenced. Multiples of known designs are included; although their existence is trivially implied, information concerning their number and resolvability usually is not.

The admissible parameter sets of nontrivial BIBDs satisfying r ≤ 41, 3 ≤ kv/2 and conditions (1) and (2) of Proposition 1.2 are ordered lexicographically by r, k, and λ (in this order). The column “Nd” contains the number Nd(v, b, r, k, λ) of pairwise nonisomorphic BIBD(v, b, r, k, λ) or the best known lower bound for this number. The column “Nr” contains a dash (-) if condition (3) of Proposition 1.10 is not satisfied. Otherwise it contains the number Nr of pairwise nonisomorphic resolutions of BIBD(v, b, r, k, λ)’s or the best known lower bound. The number of nonisomorphic RBIBDs is not necessarily Nr. Indeed, there are seven nonisomorphic resolutions of BIBD(15,35,7,3,1)s but only four nonisomorphic RBIBD(15,35,7,3,1)s (see Example II.2.76). The symbol ? indicates that the existence of the corresponding BIBD (RBIBD, respectively) is in doubt. The meanings of the “Comments” are:

 m#x m-multiple of an existing BIBD #x m#x* m-multiple of #x that does not exist or whose existence is undecided R#x (D#x) residual (derived) design of #x that exists R#x* (D#x*) residual (derived) design of #x that does not exist or whose existence is undecided #x+#y union of two designs on the same set of elements #x↓#y design #x is a design $( V , B )$ with parameters (v, b, r, k, λ); design #y is a design $( V , D )$ with parameters (v, b′, b − r, k + 1, r − λ); add a new point ∞ to each block of $B$ to obtain $B ^$ ; then $( V , B ^ ∪ D )$ is a design with parameters (v + 1, b + b′, b, k + 1, r). PG (AG) projective (affine) geometry (see §VII.2) ×1 BIBD does not exist by Bruck–Ryser–Chowla (BRC) Theorem (see §II.6.2) ×2 BIBD is a residual of a BIBD that does not exist by the BRC theorem, and λ = 1 or 2 ×3 RBIBD does not exist by Bose’s condition (see Theorem II.7.28). HD RBIBD(4t, 8t − 2, 4t − 1, 2t, 2t − 1) exists from a symmetric (Hadamard) BIBD(4t − 1, 4t − 1, 2t − 1, 2t − 1, t − 1); see §V.1.

Typically no references are given under “Ref” for multiple, derived, or residual designs of known BIBDs. A trivial formula giving Nd(v, mb, mr, k, mλ) ≥ n + 1 is often used provided Nd(v, b, r, k, λ) ≥ n, m ≥ 2, n ≥ 1 (similarly for Nr). The column “Where?” gives a pointer to an explicit construction.

No

v

b

r

k

λ

Nd

Where?

1

7

7

3

3

1

1

- PG(2,2)

II.6.4

2

9

12

4

3

1

1

1 R#3,AG(2,3)

1.22

3

13

13

4

4

1

1

- PG(2,3)

1.26

4

6

10

5

3

2

1

0 R#7,×3

1.18

5

16

20

5

4

1

1

1 R#6,AG(2,4)

1.31

6

21

21

5

5

1

1

- PG(2,4)

VI.18.73

7

11

11

5

5

2

1

-

1.26

8

13

26

6

3

1

2

- [1544]

1.27

9

7

14

6

3

2

4

- 2#1,D#20 [1665]

1.19

10

10

15

6

4

2

3

- R#13 [1665]

1.25

11

25

30

6

5

1

1

1 R#12,AG(2,5)

12

31

31

6

6

1

1

- PG(2,5)

VI.18.73

13

16

16

6

6

2

3

- [898]

1.32

14

15

35

7

3

1

80

7 PG(3,2) [1241, 1544]

1.28

15

8

14

7

4

3

4

1 R#20,AG2 (3,2) [1241, 898]

1.21

16

15

21

7

5

2

0

0 R#19*,×2

17

36

42

7

6

1

0

0 R#18*,×2,AG(2,6)

18

43

43

7

7

1

0

- × 1,PG(2,6)

19

22

22

7

7

2

0

- ×1

20

15

15

7

7

3

5

- PG2 (3, 2) [898]

1.30

21

9

24

8

3

2

36

9 2#2,D#40 [1547]

1.23

22

25

50

8

4

1

18

- [1339, 1945]

1.34

23

13

26

8

4

2

2461

- 2#3 [1743]

24

9

18

8

4

3

11

- D#41 [898]

1.24

25

21

28

8

6

2

0

- R#28*, ×2

26

49

56

8

7

1

1

1 R#27,AG(2,7)

27

57

57

8

8

1

1

- PG(2,7)

VI.18.73

28

29

29

8

8

2

0

-×1

29

19

57

9

3

1

11084874829

- [1268]

VI.16.12

30

10

30

9

3

2

960

- D#54 [537, 856, 1174]

VI.16.81

31

7

21

9

3

3

10

- 3#1 [696]

1.20

32

28

63

9

4

1

≥ 4747

≥ 7 [1346, 1533, 1548]

III.1.8

33

10

18

9

5

4

21

0 R#41,×3 [898]

VI.16.85

34

46

69

9

6

1

0

- [1138]

35

16

24

9

6

3

18920

- R#40 [1938]

II.6.30

36

28

36

9

7

2

8

0 R#39,×3 [124]

37

64

72

9

8

1

1

1 R#38,AG(2,8)

38

73

73

9

9

1

1

- PG(2,8)

VI.18.73

39

37

37

9

9

2

4

- [124]

II.6.47

40

25

25

9

9

3

78

- [694]

II.6.47

41

19

19

9

9

4

6

- [898]

1.33

42

21

70

10

3

1

≥ 62336617

≥ 63745 [518, 1263]

VI.16.12

43

6

20

10

3

4

4

1 2#4 [1241, 976]

44

16

40

10

4

2

≥ 2.2 · 106

339592 2#5 [696, 1267]

45

41

82

10

5

1

≥ 15

- [1347]

VI.16.16

46

21

42

10

5

2

≥ 22998

- 2#6 [2064]

47

11

22

10

5

4

4393

- 2#7,D#63 [323]

48

51

85

10

6

1

-

49

21

30

10

7

3

3809

0 R#54,×3 [1016, 1241, 1946]

50

36

45

10

8

2

0

- R#53*,×2

51

81

90

10

9

1

7

7 R#52,AG(2,9) [679, 1373]

52

91

91

10

10

1

4

- PG(2,9) [679, 1373]

VI.18.73

53

46

46

10

10

2

0

- ×1

54

31

31

10

10

3

151

- [1941]

II.6.47

55

12

44

11

3

2

242995846

74700 D#84 [1704]

VI.16.81

56

12

33

11

4

3

≥ 17172470

5 D#85* [1632, 1938]

VI.16.83

57

45

99

11

5

1

≥ 16

? [1548]

VI.16.31

58

12

22

11

6

5

11603

1 R#63,HD [1016, 1241, 1743]

59

45

55

11

9

2

≥ 16

0 R#62,×3 [1241, 692]

60

100

110

11

10

1

0

0 R#61*,AG(2,10) [1378]

61

111

111

11

11

1

0

- PG(2,10) [1378]

62

56

56

11

11

2

≥ 5

- [1188]

II.6.47

63

23

23

11

11

5

1106

- [1016, 1172, 1943]

VI.18.73

64

25

100

12

3

1

≥ 1014

- [1544]

VI.16.12

65

13

52

12

3

2

≥ 1897386

- 2#8,D#96 [790]

66

9

36

12

3

3

22521

426 3#2 [1543, 1707]

67

7

28

12

3

4

35

- 4#1 [976]

68

37

111

12

4

1

≥ 51402

- [587, 1349]

VI.16.14

69

19

57

12

4

2

≥ 423

- [1533]

VI.16.15

70

13

39

12

4

3

≥ 3702

- 3#3,D#97 [1548, 1940]

71

10

30

12

4

4

13769944

- 2#10 [696]

72

25

60

12

5

2

≥ 118884

≥ 748 2#11 [1225, 2062]

73

61

122

12

6

1

?

-

74

31

62

12

6

2

≥ 72

- 2#12 [1239]

75

21

42

12

6

3

≥ 236

- [1262]

VI.16.18

76

16

32

12

6

4

≥ 111

- 2#13 [1224]

77

13

26

12

6

5

19072802

- D#98 [1016, 1267]

78

22

33

12

8

4

0

- R#85* [1372]

79

33

44

12

9

3

≥ 3375

- R#84 [1551, 1999]

80

55

66

12

10

2

0

- R#83*,×2

81

121

132

12

11

1

≥ 1

≥ 1 R#82,AG(2,11)

82

133

133

12

12

1

≥ 1

- PG(2,11)

VI.18.73

83

67

67

12

12

2

0

- ×1

84

45

45

12

12

3

≥ 3752

- [1551]

VI.18.73

85

34

34

12

12

4

0

- ×1

86

27

117

13

3

1

≥ 1011

≥ 1.4 · 1013 AG(3,3) [1546, 2148]

VI.16.12

87

40

130

13

4

1

≥ 106

≥ 2 PG(3,3) [1241]

VI.16.14

88

66

143

13

6

1

≥ 1

? [693]

II.3.32

89

14

26

13

7

6

15111019

0 R#98,×3 [1267, 1999]

90

27

39

13

9

4

≥ 2.45 · 108

68 R#97,AG2 (3, 3) [1241, 1380, 1381]

91

40

52

13

10

3

?

0 R#96*,×3

92

66

78

13

11

2

≥ 2

0 R#95,×3 [1241, 114]

93

144

156

13

12

1

?

? R#94*,AG(2,12)

94

157

157

13

13

1

?

- PG(2,12)

95

79

79

13

13

2

≥ 2

- [114]

II.6.47

96

53

53

13

13

3

0

- ×1

97

40

40

13

13

4

≥ 1108800

- PG2(3, 3) [1374]

VI.18.73

98

27

27

13

13

6

208310

- [1943]

VI.18.73

99

15

70

14

3

2

≥ 685521

≥ 36 2#14,D#140 [734, 1548]

100

22

77

14

4

2

≥ 7921

- [827]

VI.16.15

101

8

28

14

4

6

2310

4 2#15 [1743]

102

15

42

14

5

4

≥ 896

0 2#16*,D#141* [1265, 2031]

VI.16.85

103

36

84

14

6

2

≥ 5

≥ 2 2#17* [1228, 2130]

VI.16.86

104

15

35

14

6

5

≥ 117

- D#142 [1016, 1533]

VI.16.86

105

85

170

14

7

1

?

-

106

43

86

14

7

2

≥ 4

- 2#18* [1]

VI.16.30

107

29

58

14

7

3

≥ 1

-

VI.16.30

108

22

44

14

7

4

≥ 3393

- 2#19* [2031]

VI.16.30

109

15

30

14

7

6

≥ 57810

- 2#20,D#143 [1545]

110

78

91

14

12

2

0

- R#113*,×2

111

169

182

14

13

1

≥ 1

≥ 1 R#112,AG(2,13)

112

183

183

14

14

1

≥ 1

- PG(2,13)

VI.18.73

113

92

92

14

14

2

0

- ×1

114

31

155

15

3

1

≥ 6 · 1016

- [1548]

VI.16.12

115

16

80

15

3

2

≥ 1013

- D#169 [1548]

VI.16.13

116

11

55

15

3

3

≥ 436800

- [1548]

VI.16.24

117

7

35

15

3

5

109

- 5#1 [1938]

118

6

30

15

3

6

6

0 3#4 [976, 1548]

119

16

60

15

4

3

≥ 6 · 105

≥ 6 · 105 3#5,D#170 [1548]

120

61

183

15

5

1

≥ 10

- [587]

VI.16.16

121

31

93

15

5

2

≥ 1

-

VI.16.17

122

21

63

15

5

3

≥ 109

- 3#6 [331]

123

16

48

15

5

4

≥ 294

- D#171 [352, 734, 1016]

124

13

39

15

5

5

≥ 76

- [1016, 2059, 734]

VI.16.17

125

11

33

15

5

6

≥ 127

- 3#7 [324, 1224]

126

76

190

15

6

1

≥ 1

- [1612]

II.3.32

127

26

65

15

6

3

≥ 1

- [1016]

VI.16.89

128

16

40

15

6

5

≥ 25

- D#172 [1016, 1533, 734]

129

91

195

15

7

1

≥ 2

? [246]

VI.16.70

130

16

30

15

8

7

≥ 9 · 107

5 R#143,AG3(4, 2),HD [1271, 1319]

131

21

35

15

9

6

≥ 104

- R#142 [1016, 404]

132

136

204

15

10

1

?

-

133

46

69

15

10

3

?

-

134

28

42

15

10

5

≥ 3

- R#141* [2090]

135

56

70

15

12

3

≥ 4

- R#140 [1005]

136

91

105

15

13

2

0

0 R#139*,×2

137

196

210

15

14

1

0

0 R#138*,×2,AG(2,14)

138

211

211

15

15

1

0

- × 1,PG(2,14)

139

106

106

15

15

2

0

-×1

140

71

71

15

15

3

≥ 72

- [1005, 1832]

II.6.47

141

43

43

15

15

5

0

-×1

142

36

36

15

15

6

≥ 25634

- [1944]

VI.18.73

143

31

31

15

15

7

≥ 22478260

- PG3(4, 2) [1375]

VI.18.80

144

33

176

16

3

1

≥ 1013

≥ 4494390 [1548, 562]

VI.16.12

145

9

48

16

3

4

16585031

149041 4#2 [1707]

146

49

196

16

4

1

≥ 769

- [396, 978, 587]

VI.16.14

147

25

100

16

4

2

≥ 17

- 2#22

148

17

68

16

4

3

≥ 542

- D#185 [1999, 734]

149

13

52

16

4

4

≥ 2462

-4#3

150

9

36

16

4

6

270474142

- 2#24 [1705]

151

65

208

16

5

1

≥ 2

≥ 1 [514, 587]

VI.16.16

152

81

216

16

6

1

?

-

153

21

56

16

6

4

≥ 1

- 2#25* [1042]

II.3.32

154

49

112

16

7

2

≥ 1

≥ 1 2#26

155

113

226

16

8

1

?

-

156

57

114

16

8

2

≥ 1362

- 2#27 [1239]

157

29

58

16

8

4

≥ 2

- 2#28* [1999]

VI.16.30

158

17

34

16

8

7

≥ 28

- D#186 [734, 1999, 2060]

159

145

232

16

10

1

?

-

160

25

40

16

10

6

≥ 43

- R#172 [1236, 1938]

161

33

48

16

11

5

≥ 19

0 R#171,×3 [1241, 352]

162

177

236

16

12

1

?

-

163

45

60

16

12

4

≥ 1

- R#170 [180]

164

65

80

16

13

3

?

0 R#169*,×3

165

105

120

16

14

2

?

- R#168*

166

225

240

16

15

1

?

? R#167*,AG(2,15)

167

241

241

16

16

1

?

- PG(2,15)

168

121

121

16

16

2

?

-

169

81

81

16

16

3

?

-

170

61

61

16

16

4

≥ 6

- [1398]

II.6.47

171

49

49

16

16

5

≥ 12146

- [1321]

II.6.47

172

41

41

16

16

6

≥ 115307

- [1938]

II.6.47

173

18

102

17

3

2

≥ 4 · 1014

≥ 173 D#217* [734, 1040, 1548]

VI.16.81

174

52

221

17

4

1

≥ 206

≥ 30 [392, 587, 1377]

VI.16.14

175

35

119

17

5

2

≥ 1

≥ 1 [1999, 1]

VI.16.17

176

18

51

17

6

5

≥ 582

≥ 2 D#218* [1999, 1243, 734]

VI.16.86

177

35

85

17

7

3

≥ 2

? [1, 1044, 1548]

178

120

255

17

8

1

≥ 94

≥ 1 [1729, 1877]

179

18

34

17

9

8

≥ 103

0 R#186,×3 [404, 1241, 1999]

180

52

68

17

13

4

≥ 6

0 R#185,×3 [1241, 2065]

181

120

136

17

15

2

0

0 R#184*,×2

182

256

272

17

16

1

≥ 189

≥ 189 R#183,AG(2,16) [1207, 1208]

183

273

273

17

17

1

≥ 22

- PG(2,16) [660, 1208, 1207]

VI.18.73

184

137

137

17

17

2

0

-×1

185

69

69

17

17

4

≥ 4

- [2065]

II.6.47

186

35

35

17

17

8

≥ 108131

- [1944]

VI.18.73

187

37

222

18

3

1

≥ 1010

- [1463, 1999]

VI.16.12

188

19

114

18

3

2

≥ 2 · 109

- 2#29,D#231* [1548]

189

13

78

18

3

3

≥ 3 · 109

- 3#8 [1548]

190

10

60

18

3

4

≥ 961

- 2#30

191

7

42

18

3

6

418

- 6#1 [1743, 1938]

192

28

126

18

4

2

≥ 139

≥ 8 2#32

193

10

45

18

4

6

≥ 14819

- 3#10 [1548]

194

25

90

18

5

3

≥ 1017

≥ 1017 3#11 [1548]

195

10

36

18

5

8

≥ 135922

5 2#33 [1271, 1545]

196

91

273

18

6

1

≥ 4

- [534, 1607]

VI.16.18

197

46

138

18

6

2

≥ 1

- 2#34*

VI.16.18

198

31

93

18

6

3

≥ 1022

- 3#12 [1548]

199

19

57

18

6

5

≥ 1535

- D#232* [734]

II.7.46

200

16

48

18

6

6

≥ 108

- 3#13,2#35 [1548]

201

28

72

18

7

4

≥ 392

? 2#36 [1224]

202

64

144

18

8

2

≥ 121

≥ 121 2#37 [1225]

203

145

290

18

9

1

?

-

204

73

146

18

9

2

≥ 3500

- 2#38 [1239]

205

49

98

18

9

3

≥ 1

- [1628]

VI.16.30

206

37

74

18

9

4

≥ 852

- 2#39 [1224]

207

25

50

18

9

6

≥ 79

- 2#40

208

19

38

18

9

8

≥ 108

- 2#41,D#233 [734]

209

55

99

18

10

3

?

-

210

100

150

18

12

2

?

-

211

34

51

18

12

6

≥ 2

- R#218* [1496]

212

85

102

18

15

3

?

- R#217*

213

136

153

18

16

2

?

- R#216*

214

289

306

18

17

1

≥ 1

≥ 1 R#215,AG(2,17)

215

307

307

18

18

1

≥ 1

- PG(2,17)

VI.18.73

216

154

154

18

18

2

?

-

217

103

103

18

18

3

0

-×1

218

52

52

18

18

6

0

-×1

219

39

247

19

3

1

≥ 1044

≥ 1626684 [1463, 562]

VI.16.12

220

20

95

19

4

3

≥ 10040

≥ 204 D#270 [1999, 623, 734]

VI.16.83

221

20

76

19

5

4

≥ 10067

≥ 14 D#271* [734, 1042]

VI.16.85

222

96

304

19

6

1

≥ 1

? [1609]

II.3.32

223

153

323

19

9

1

?

?

224

20

38

19

10

9

≥ 1016

3 R#233,HD [1319]

225

39

57

19

13

6

?

0 R#232*,×3

226

96

114

19

16

3

?

0 R#231*,×3

227

153

171

19

17

2

0

0 R#230*,×2

228

324

342

19

18

1

?

? R#229*,AG(2,18)

229

343

343

19

19

1

?

- PG(2,18)

230

172

172

19

19

2

0

-×1

231

115

115

19

19

3

?

-

232

58

58

19

19

6

0

-×1

233

39

39

19

19

9

≥ 5.87 · 1014

- [1374]

V.1.28

234

21

140

20

3

2

≥ 5 · 1014

≥ 79 2#42,D#307 [1548]

235

9

60

20

3

5

5862121434

203047732 5#2 [1707]

236

6

40

20

3

8

13

1 4#4 [1174]

237

61

305

20

4

1

≥ 18132

- [1999, 587]

VI.16.61

238

31

155

20

4

2

≥ 43

- [1999, 734]

VI.16.15

239

21

105

20

4

3

≥ 26320

- D#308* [1999, 734]

VI.16.15

240

16

80

20

4

4

≥ 6 · 105

≥ 6 · 105 4#5 [1548]

241

13

65

20

4

5

≥ 103

- 5#3 [1548]

242

11

55

20

4

6

≥ 348

- [734]

VI.16.15

243

81

324

20

5

1

≥ 1

- [1999]

VI.16.16

244

41

164

20

5

2

≥ 6

-2#45

245

21

84

20

5

4

≥ 109

- 4#6,D#309 [1548]

II.7.46

246

17

68

20

5

5

≥ 7260

- [514, 734]

VI.16.17

247

11

44

20

5

8

≥ 4394

-4#7

248

51

170

20

6

2

≥ 446

- 2#48* [2063]

VI.16.91

249

21

70

20

6

5

≥ 1

- D#310 [1042]

VI.16.18

250

21

60

20

7

6

≥ 3810

≥ 1 2#49,D#311* [1]

251

36

90

20

8

4

≥ 2

- 2#50* [1, 2071]

252

81

180

20

9

2

≥ 1169

≥ 1169 2#51 [1225]

253

181

362

20

10

1

?

-

254

91

182

20

10

2

≥ 46790

- 2#52 [1239]

255

61

122

20

10

3

≥ 1

- [1628]

VI.16.30

256

46

92

20

10

4

≥ 1

- 2#53* [1628]

VI.16.30

257

37

74

20

10

5

≥ 1

- [1999]

VI.16.30

258

31

62

20

10

6

≥ 152

- 2#54

259

21

42

20

10

9

≥ 4

- D#312 [1999, 248]

260

111

185

20

12

2

?

-

261

45

75

20

12

5

?

-

262

141

188

20

15

2

?

-

263

57

76

20

15

5

?

- R#271*

264

36

48

20

15

8

≥ 1

- [1912]

265

76

95

20

16

4

≥ 1

- R#270 [1999, 514]

266

171

190

20

18

2

?

- R#269*

267

361

380

20

19

1

≥ 1

≥ 1 R#268,AG(2,19)

268

381

381

20

20

1

≥ 1

- PG(2,19)

VI.18.73

269

191

191

20

20

2

?

-

270

96

96

20

20

4

≥ 2

- [1839]

VI.18.73

271

77

77

20

20

5

0

-×1

272

43

301

21

3

1

≥ 5 · 1064

- [1042, 1463]

VI.16.12

273

22

154

21

3

2

≥ 3 · 109

- D#336* [1548]

VI.16.81

274

15

105

21

3

3

≥ 1015

≥ 1011 3#14 [1548]

275

8

56

21

3

6

3077244

- [1938]

VI.16.82

276

7

49

21

3

7

1508

- 7#1 [1743, 1938]

277

64

336

21

4

1

≥ 1.4 · 1031

2.5 · 1037 [1546]

VI.16.14

278

8

42

21

4

9

8360901

10 3#15 [696]

279

85

357

21

5

1

≥ 3.2 · 1038

≥ 1 PG(3,4) [1546, 587]

VI.16.70

280

15

63

21

5

6

≥ 2211

≥ 149 3#16* [1549]

VI.16.17

281

106

371

21

6

1

≥ 1

- [1606]

II.3.32

282

36

126

21

6

3

≥ 1

≥ 1 3#17* [1042, 1]

VI.16.86

283

22

77

21

6

5

≥ 3

- D#337,#6|#153 [1053]

VI.16.18

284

16

56

21

6

7

≥ 1

- #13+#128 [1903]

285

127

381

21

7

1

?

-

286

64

192

21

7

2

≥ 1

- [2]

II.3.32

287

43

129

21

7

3

≥ 1

- 3#18* [2144]

VI.16.30

288

22

66

21

7

6

≥ 1

- 3#19*,D#338* [1042]

II.7.47

289

19

57

21

7

7

≥ 1

- [2144]

II.7.46

290

15

45

21

7

9

≥ 108

- 3#20 [1548]

291

57

133

21

9

3

≥ 1

- [21]

292

190

399

21

10

1

?

?

293

22

42

21

11

10

≥ 2

0 R#312,×3 [1241, 1999, 1013]

294

232

406

21

12

1

?

-

295

274

411

21

14

1

?

-

296

92

138

21

14

3

?

-

297

40

60

21

14

7

?

- R#311*

298

295

413

21

15

1

?

-

299

50

70

21

15

6

≥ 1

- R#310 [1185]

300

64

84

21

16

5

≥ 10810800

≥ 157 R#309,AG2(3,4) [1374, 1223]

301

85

105

21

17

4

?

0 R#308*,×3

302

120

140

21

18

3

?

- R#307*

303

190

210

21

19

2

?

0 R#306*,×3

304

400

420

21

20

1

?

? R#305*,AG(2,20)

305

421

421

21

21

1

?

- PG(2,20)

306

211

211

21

21

2

?

-

307

141

141

21

21

3

0

- ×1

308

106

106

21

21

4

0

- ×1

309

85

85

21

21

5

≥ 213964

- PG2 (3,4) [1223]

VI.18.73

310

71

71

21

21

6

≥ 2

- [1185]

II.6.47

311

61

61

21

21

7

0

-×1

312

43

43

21

21

10

≥ 82

- [1013, 2066]

VI.18.73

313

45

330

22

3

1

≥ 6 · 1076

≥ 84 [1042, 1463, 1548]

VI.16.12

314

12

88

22

3

4

≥ 20476

≥ 3 2#55

315

34

187

22

4

2

≥ 1

- [1042]

VI.16.15

316

12

66

22

4

6

≥ 1.7 · 106

≥ 1 2#56

317

45

198

22

5

2

≥ 17

? 2#57

318

111

407

22

6

1

≥ 1

- [1609]

II.3.32

319

12

44

22

6

10

≥ 11604

545 2#58 [324, 1271]

320

133

418

22

7

1

?

?

321

45

110

22

9

4

≥ 1353

? 2#59 [1224]

322

100

220

22

10

2

≥ 1

? 2#60* [528]

IV.2.67

323

221

442

22

11

1

?

-

324

111

222

22

11

2

?

- 2#61*

325

56

112

22

11

4

≥ 2696

- 2#62 [1224]

326

45

90

22

11

5

≥ 1

- [1628]

VI.16.30

327

23

46

22

11

10

≥ 1103

- 2#63,D#351

328

287

451

22

14

1

?

-

329

45

66

22

15

7

?

? R#338*

330

56

77

22

16

6

≥ 3

- R#337 [2045]

331

133

154

22

19

3

?

0 R#336*,×3

332

210

231

22

20

2

0

- R#335*,×2

333

441

462

22

21

1

0

0 R#334*,×2,AG(2,21)

334

463

463

22

22

1

0

- ×1,PG(2,21)

335

232

232

22

22

2

0

-×1

336

155

155

22

22

3

?

-

337

78

78

22

22

6

≥ 3

- [2045]

II.6.47

338

67

67

22

22

7

0

-×1

339

24

184

23

3

2

≥ 3 · 109

≥ 1 D#404* [1040, 1548]

VI.16.81

340

24

138

23

4

3

≥ 1

≥ 1 D#405* [144, 1042]

VI.16.83

341

24

92

23

6

5

≥ 1

≥ 1 D#406* [848, 1042]

VI.16.86

342

70

230

23

7

2

≥ 1

? [2]

VI.16.88

343

24

69

23

8

7

≥ 1

? D#407 [1042]

VI.16.88

344

70

161

23

10

3

?

?

345

231

483

23

11

1

?

?

346

24

46

23

12

11

≥ 1027

130 R#351,HD [1271, 1375, 1999]

347

231

253

23

21

2

0

0 R#350*,×2

348

484

506

23

22

1

0

0 R#349*,×2,AG(2,22)

349

507

507

23

23

1

0

- ×1,PG(2,22)

350

254

254

23

23

2

0

-×1

351

47

47

23

23

11

≥ 55

- [620]

VI.18.73

352

49

392

24

3

1

≥ 6 · 1014

- [1042, 1463]

VI.16.12

353

25

200

24

3

2

≥ 1014

- 2#64,D#438*

354

17

136

24

3

3

≥ 4968

- [1548]

VI.16.24

355

13

104

24

3

4

≥ 108

- 4#8 [1548]

356

9

72

24

3

6

≥ 107

≥ 105 6#2 [1548]

357

7

56

24

3

8

5413

- 8#1 [1743, 1938]

358

73

438

24

4

1

≥ 107

- [332]

VI.16.61

359

37

222

24

4

2

≥ 4

- 2#68

360

25

150

24

4

3

≥ 1022

- 3#22,D#439* [1548]

361

19

114

24

4

4

≥ 424

- 2#69

362

13

78

24

4

6

≥ 108

- 6#3 [1548]

363

10

60

24

4

8

≥ 1759614

- 4#10 [1545]

364

9

54

24

4

9

≥ 106

- 3#24 [1548]

365

25

120

24

5

4

≥ 1017

≥ 1017 4#11,D#440 [1548]

366

121

484

24

6

1

≥ 1

- [1042, 1607]

VI.16.31

367

61

244

24

6

2

≥ 1

- 2#73*

VI.16.18

368

41

164

24

6

3

≥ 1

-

VI.16.18

369

31

124

24

6

4

≥ 1022

- 4#12 [1548]

370

25

100

24

6

5

≥ 1

- D#441 [1042]

371

21

84

24

6

6

≥ 1

- 3#25*,2#75

372

16

64

24

6

8

≥ 108

- 4#13 [1548]

373

13

52

24

6

10

≥ 2572157

- 2#77

374

49

168

24

7

3

≥ 1052

≥ 1052 3#26 [1548]

375

169

507

24

8

1

?

-

376

85

255

24

8

2

≥ 1

- [1628]

VI.16.30

377

57

171

24

8

3

≥ 1063

- 3#27 [1548]

378

43

129

24

8

4

≥ 1

- [2144]

VI.16.30

379

29

87

24

8

6

≥ 1

- 3#28* [1042]

VI.16.30

380

25

75

24

8

7

≥ 1

- D#442* [2144]

VI.16.31

381

22

66

24

8

8

≥ 1

- 2#78*

VI.16.30

382

33

88

24

9

6

≥ 3376

- 2#79

383

55

132

24

10

4

?

- 2#80*

384

25

60

24

10

9

≥ 3

- D#443 [607, 2144]

385

121

264

24

11

2

≥ 365

≥ 365 2#81 [1225]

386

265

530

24

12

1

?

-

387

133

266

24

12

2

≥ 9979200

- 2#82 [1239]

388

89

178

24

12

3

?

-

389

67

134

24

12

4

≥ 1

- 2#83* [1]

VI.16.30

390

45

90

24

12

6

≥ 3753

- 2#84

391

34

68

24

12

8

≥ 1

- 2#85* [1]

VI.16.30

392

25

50

24

12

11

≥ 10

- D#444 [607, 2144]

393

105

180

24

14

3

?

-

394

85

136

24

15

4

?

-

395

46

69

24

16

8

≥ 1

- R#407 [1185]

396

69

92

24

18

6

?

- R#406*

397

115

138

24

20

4

?

- R#405*

398

161

184

24

21

3

?

- R#404*

399

49

56

24

21

10

≥ 1

- [2144]

VI.16.32

400

253

276

24

22

2

0

- R#403*,×2

401

529

552

24

23

1

≥ 1

≥ 1 R#402,AG(2,23)

402

553

553

24

24

1

≥ 1

- PG(2,23)

VI.18.73

403

277

277

24

24

2

0

-×1

404

185

185

24

24

3

0

-×1

405

139

139

24

24

4

?

-

406

93

93

24

24

6

0

-×1

407

70

70

24

24

8

≥ 28

- [1185, 620]

II.6.47

408

51

425

25

3

1

≥ 6 · 1053

≥ 9419 [1463, 1976]

VI.16.12

409

6

50

25

3

10

19

0 5#4 [1174, 1548]

410

76

475

25

4

1

≥ 169574

≥ 1 [396, 587, 1047]

VI.16.14

411

16

100

25

4

5

≥ 106

≥ 106 5#5 [1548]

412

101

505

25

5

1

≥ 3

- [393]

VI.16.62

413

51

255

25

5

2

≥ 1

- [1042]

VI.16.17

414

26

130

25

5

4

≥ 1

- D#471* [1042]

II.7.46

415

21

105

25

5

5

≥ 109

- 5#6 [1548]

416

11

55

25

5

10

≥ 3337

- 5#7

417

126

525

25

6

1

≥ 2

≥ 1 [679, 1042, 1578]

VI.16.92

418

176

550

25

8

1

?

?

419

226

565

25

10

1

?

-

420

76

190

25

10

3

?

-

421

46

115

25

10

5

≥ 1

- [2071]

422

26

65

25

10

9

≥ 19

- D#472 [1042, 1556]

423

276

575

25

12

1

?

?

424

26

50

25

13

12

≥ 1

0 R#444,×3 [1241, 1999]

425

351

585

25

15

1

?

-

426

51

85

25

15

7

?

-

427

36

60

25

15

10

≥ 1

- R#443 [1042]

428

51

75

25

17

8

?

0 R#442*,×3

429

76

100

25

19

6

≥ 1

0 R#441,×3

430

476

595

25

20

1

?

-

431

96

120

25

20

5

≥ 1

- R#440

432

126

150

25

21

4

?

0 R#439*,×3

433

176

200

25

22

3

?

0 R#438*,×3

434

276

300

25

23

2

?

0 R#437*,×3

435

576

600

25

24

1

?

? R#436*,AG(2,24)

436

601

601

25

25

1

?

- PG(2,24)

437

301

301

25

25

2

?

-

438

201

201

25

25

3

?

-

439

151

151

25

25

4

0

-×1

440

121

121

25

25

5

≥ 1

- [1773]

II.6.47

441

101

101

25

25

6

≥ 1

-

VI.18.73

442

76

76

25

25

8

0

-×1

443

61

61

25

25

10

≥ 24

- [1720]

II.6.47

444

51

51

25

25

12

≥ 1

- [1999]

V.1.39

445

27

234

26

3

2

≥ 1011

≥ 910 2#86,D#511*

446

40

260

26

4

2

≥ 106

≥ 1 2#87

447

14

91

26

4

6

≥ 4

- [1042, 1588]

II.5.29

448

105

546

26

5

1

≥ 1

≥ 1 [41]

IV.2.2

449

66

286

26

6

2

≥ 1

≥ 1 2#88 [958]

450

27

117

26

6

5

≥ 1

- D#512* [1042]

VI.16.86

451

14

52

26

7

12

≥ 1363846

1363486 2#89 [1264]

452

92

299

26

8

2

≥ 1

- [19]

453

27

78

26

9

8

≥ 8072

≥ 13 2#90,D#513

454

235

611

26

10

1

?

-

455

40

104

26

10

6

≥ 1

? 2#91* [1]

456

66

156

26

11

4

≥ 494

? 2#92 [1224]

457

144

312

26

12

2

≥ 1

? 2#93* [528]

IV.2.67

458

313

626

26

13

1

?

-

459

157

314

26

13

2

?

- 2#94*

460

105

210

26

13

3

?

-

461

79

158

26

13

4

≥ 940

- 2#95 [1224]

462

53

106

26

13

6

≥ 1

- 2#96* [2144]

VI.16.30

463

40

80

26

13

8

≥ 390

- 2#97

464

27

54

26

13

12

≥ 208311

- 2#98,D#514

465

40

65

26

16

10

≥ 1

- R#472 [2067]

466

105

130

26

21

5

?

0 R#471*,×3

467

300

325

26

24

2

0

- R#470*,×2

468

625

650

26

25

1

≥ 33

≥ 33 R#469,AG(2,25) [1251]

469

651

651

26

26

1

≥ 17

- PG(2,25) [1251]

VI.18.73

470

326

326

26

26

2

0

- ×1

471

131

131

26

26

5

?

-

472

66

66

26

26

10

≥ 588

- [2067, 1720]

II.6.47

473

55

495

27

3

1

≥ 6 · 1076

- [1463]

VI.16.12

474

28

252

27

3

2

≥ 1055

- D#564* [1548]

VI.16.13

475

19

171

27

3

3

≥ 1017

- 3#29 [1548]

476

10

90

27

3

6

≥ 1012

- 3#30 [1548]

477

7

63

27

3

9

17785

- 9#1 [1743, 1938]

478

28

189

27

4

3

≥ 1032

≥ 1026 3#32,D#565* [1548]

479

55

297

27

5

2

≥ 1

≥ 1 [1042, 848]

IV.2.67

480

10

54

27

5

12

≥ 108

0 3#33 [1548]

481

136

612

27

6

1

≥ 1

- [1610]

II.3.32

482

46

207

27

6

3

≥ 1028

- 3#34* [1042, 416]

VI.16.90

483

28

126

27

6

5

≥ 2

- D#566* [2130]

VI.16.18

484

16

72

27

6

9

≥ 18921

- 3#35

485

28

108

27

7

6

≥ 5047

≥ 1 3#36,D#567 [848, 1224]

VI.16.87

486

64

216

27

8

3

≥ 1077

≥ 1077 3#37 [1548]

487

217

651

27

9

1

?

-

488

109

327

27

9

2

≥ 1

- [1548]

VI.16.57

489

73

219

27

9

3

≥ 1090

- 3#38 [1548]

490

55

165

27

9

4

≥ 1

-[1]

VI.16.30

491

37

111

27

9

6

≥ 1037

- 3#39 [1548]

492

28

84

27

9

8

≥ 3

- D#568* [1548, 2130]

II.7.48

493

25

75

27

9

9

≥ 1028

- 3#40 [1548]

494

19

57

27

9

12

≥ 1016

- 3#41 [1548]

495

55

135

27

11

5

?

?

496

100

225

27

12

3

?

-

497

28

63

27

12

11

≥ 246

- D#569 [1236, 1379, 710]

498

325

675

27

13

1

?

?

499

28

54

27

14

13

≥ 9 · 1021

7570 R#514,HD [1271, 1319, 2039]

500

190

342

27

15

2

?

-

501

55

99

27

15

7

?

-

502

460

690

27

18

1

?

-

503

154

231

27

18

3

?

-

504

52

78

27

18

9

≥ 2

- R#513 [957]

505

91

117

27

21

6

?

- R#512*

506

208

234

27

24

3

?

- R#511*

507

325

351

27

25

2

?

0 R#510*,×3

508

676

702

27

26

1

?

? R#509*,AG(2,26)

509

703

703

27

27

1

?

- PG(2,26)

510

352

352

27

27

2

?

-

511

235

235

27

27

3

0

-×1

512

118

118

27

27

6

0

-×1

513

79

79

27

27

9

≥ 1463

- [1087]

II.6.47

514

55

55

27

27

13

≥ 1

- [1999, 514]

V.1.28

515

57

532

28

3

1

≥ 1090

≥ 1 [1042]

VI.16.12

516

15

140

28

3

4

≥ 1015

≥ 1011 4#14 [1548]

517

9

84

28

3

7

≥ 330

≥ 9 7#2

518

85

595

28

4

1

≥ 1015

- [1042, 332]

VI.16.14

519

43

301

28

4

2

≥ 1

-

VI.16.15

520

29

203

28

4

3

≥ 1

- D#586*

II.7.46

521

22

154

28

4

4

≥ 7922

- 2#100

522

15

105

28

4

6

≥ 31300

- [127]

VI.16.15

523

13

91

28

4

7

≥ 108

- 7#3 [1548]

524

8

56

28

4

12

≥ 2310

31 4#15 [1174, 1548]

525

15

84

28

5

8

≥ 104

≥ 1 4#16*,2#102 [1]

526

141

658

28

6

1

≥ 1

- [1605]

527

36

168

28

6

4

≥ 5

≥ 3 4#17*,2#103

528

21

98

28

6

7

≥ 1

- #75+#153

529

15

70

28

6

10

≥ 118

- 2#104

530

169

676

28

7

1

≥ 1

- [1044, 1548]

VI.16.65

531

85

340

28

7

2

≥ 2

- 2#105* [1, 959]

VI.16.30

532

57

228

28

7

3

≥ 1

- [1042]

VI.16.30

533

43

172

28

7

4

≥ 4

- 4#18*,2#106

534

29

116

28

7

6

≥ 1

- 2#107,D#587*

535

25

100

28

7

7

≥ 1

- [2144]

VI.16.31

536

22

88

28

7

8

≥ 35

- 4#19*,2#108

537

15

60

28

7

12

- 4#20 [1548]

538

50

175

28

8

4

≥ 1

-[1]

VI.16.89

539

225

700

28

9

1

?

?

540

85

238

28

10

3

?

-

541

309

721

28

12

1

?

-

542

78

182

28

12

4

?

- 2#110*

543

45

105

28

12

7

≥ 1

- #84+#163

544

169

364

28

13

2

≥ 765

≥ 765 2#111 [1225]

545

365

730

28

14

1

?

-

546

183

366

28

14

2

≥ 109

- 2#112 [1239]

547

92

184

28

14

4

?

- 2#113*

548

53

106

28

14

7

≥ 1

- [2144]

VI.16.30

549

29

58

28

14

13

≥ 1

- D#588 [2144]

550

36

63

28

16

12

≥ 8784

- R#569 [1236, 710]

551

477

742

28

18

1

?

-

552

57

84

28

19

9

?

0 R#568*,×3

553

561

748

28

21

1

?

-

554

141

188

28

21

4

?

-

555

81

108

28

21

7

≥ 1

- R#567

556

57

76

28

21

10

?

-

557

99

126

28

22

6

?

- R#566*

558

162

189

28

24

4

?

- R#565*

559

225

252

28

25

3

?

0 R#564*,×3

560

351

378

28

26

2

0

- R#563*,×2

561

729

756

28

27

1

≥ 7

≥ 7 R#562,AG(2,27) [1251]

562

757

757

28

28

1

≥ 3

- PG(2,27) [679]

VI.18.73

563

379

379

28

28

2

0

-×1

564

253

253

28

28

3

?

-

565

190

190

28

28

4

0

-×1

566

127

127

28

28

6

?

-

567

109

109

28

28

7

≥ 1

-

VI.18.73

568

85

85

28

28

9

?

-

569

64

64

28

28

12

≥ 8784

- [710]

VI.18.73

570

30

290

29

3

2

≥ 2 · 1051

≥ 1 D#655* [1040, 1548]

VI.16.81

571

88

638

29

4

1

≥ 2

≥ 1 [1042, 332, 1047]

IV.2.2

572

30

174

29

5

4

≥ 1

≥ 4 D#656 [1042, 623]

VI.16.85

573

30

145

29

6

5

≥ 1

≥ 1 D#657* [1042, 33]

VI.16.86

574

175

725

29

7

1

≥ 1

? [1184]

VI.16.31

575

117

377

29

9

2

≥ 1

? [1034]

VI.16.70

576

30

87

29

10

9

≥ 1

? D#658* [1]

VI.16.88

577

117

261

29

13

3

?

?

578

378

783

29

14

1

?

?

579

30

58

29

15

14

≥ 1

0 R#588,×3 [1241, 2144]

580

88

116

29

22

7

?

0 R#587*,×3

581

175

203

29

25

4

?

0 R#586*,×3

582

378

406

29

27

2

?

0 R#585*,×3

583

784

812

29

28

1

?

? R#584*,AG(2,28)

584

813

813

29

29

1

?

- PG(2,28)

585

407

407

29

29

2

?

-

586

204

204

29

29

4

?

-

587

117

117

29

29

7

0

-×1

588

59

59

29

29

14

≥ 1

-

VI.18.73

589

61

610

30

3

1

≥ 2 · 1024

- [1042, 1463]

VI.16.12

590

31

310

30

3

2

≥ 6 · 1016

- 2#114,D#677*

591

21

210

30

3

3

≥ 1024

≥ 1021 3#42 [1548]

592

16

160

30

3

4

≥ 1013

- 2#115

593

13

130

30

3

5

≥ 108

- 5#8 [1548]

594

11

110

30

3

6

≥ 436801

- 2#116

595

7

70

30

3

10

54613

- 10#1 [1545]

596

6

60

30

3

12

34

1 6#4,3#43 [1174, 1548]

597

46

345

30

4

2

≥ 1

- [1042]

VI.16.15

598

16

120

30

4

6

≥ 1015

≥ 1015 6#5 [1548]

599

10

75

30

4

10

≥ 29638

- 5#10 [1548]

600

121

726

30

5

1

≥ 1

- [1042]

VI.16.62

601

61

366

30

5

2

≥ 11

- 2#120

602

41

246

30

5

3

≥ 1046

- 3#45

603

31

186

30

5

4

≥ 1

- 2#121,D#678*

604

25

150

30

5

5

≥ 1017

≥ 1017 5#11 [1548]

605

21

126

30

5

6

≥ 1024

- 6#6 [1548]

606

16

96

30

5

8

≥ 12

- 2#123

607

13

78

30

5

10

≥ 31

- 2#124

608

11

66

30

5

12

≥ 106

- 6#7 [1548]

609

151

755

30

6

1

≥ 1

- [1042, 2144]

VI.16.54

610

76

380

30

6

2

≥ 1

- 2#126

611

51

255

30

6

3

≥ 1

- 3#48* [1042]

VI.16.18

612

31

155

30

6

5

≥ 1023

- 5#12,D#679 [1548]

613

26

130

30

6

6

≥ 1

- 2#127

614

16

80

30

6

10

≥ 108

- 5#13 [1548]

615

91

390

30

7

2

≥ 3

? 2#129

616

21

90

30

7

9

≥ 1018

≥ 1 3#49 [1]

617

36

135

30

8

6

≥ 3

- 3#50* [1, 1497, 2130]

618

16

60

30

8

14

≥ 9 · 107

≥ 6 2#130

619

81

270

30

9

3

≥ 10108

≥ 10108 3#51 [1548]

620

21

70

30

9

12

≥ 104

- 2#131

621

271

813

30

10

1

?

-

622

136

408

30

10

2

?

- 2#132*

623

91

273

30

10

3

≥ 10125

- 3#52 [1548]

624

55

165

30

10

5

≥ 2

- [1034]

VI.16.30

625

46

138

30

10

6

≥ 1

- 2#133*,3#53* [1]

626

31

93

30

10

9

≥ 152

- 3#54,D#680*

627

28

84

30

10

10

≥ 5

- 2#134* [1548, 2130]

II.7.48

628

166

415

30

12

2

?

-

629

56

140

30

12

6

≥ 5

- 2#135

630

34

85

30

12

10

≥ 1

- [1628]

VI.16.30

631

91

210

30

13

4

?

? 2#136*

632

196

420

30

14

2

?

? 2#137*

633

421

842

30

15

1

?

-

634

211

422

30

15

2

?

- 2#138*

635

141

282

30

15

3

?

-

636

106

212

30

15

4

?

- 2#139*

637

85

170

30

15

5

?

-

638

71

142

30

15

6

≥ 9

- 2#140

639

61

122

30

15

7

≥ 1

- [2144]

VI.16.30

640

43

86

30

15

10

≥ 1

- 2#141* [1042]

VI.16.30

641

36

72

30

15

12

≥ 25635

- 2#142

642

31

62

30

15

14

≥ 106

- 2#143,D#681

643

171

285

30

18

3

?

-

644

286

429

30

20

2

?

-

645

96

144

30

20

6

?

-

646

58

87

30

20

10

?

- R#658*

647

301

430

30

21

2

?

-

648

116

145

30

24

6

?

- R#657*

649

145

174

30

25

5

≥ 1

- R#656 [1839]

650

261

290

30

27

3

?

- R#655*

651

406

435

30

28

2

0

- R#654*,×2

652

841

870

30

29

1

≥ 1

≥ 1 R#653,AG(2,29)

653

871

871

30

30

1

≥ 1

- PG(2,29)

VI.18.73

654

436

436

30

30

2

0

-×1

655

291

291

30

30

3

?

-

656

175

175

30

30

5

≥ 2

- [331, 1839]

VI.18.73

657

146

146

30

30

6

0

-×1

658

88

88

30

30

10

0

-×1

659

63

651

31

3

1

≥ 1042

≥ 82160 PG(5,2) [1463, 1548]

VI.16.12

660

32

248

31

4

3

≥ 1

≥ 1 D#729* [1042, 144]

VI.16.83

661

125

775

31

5

1

≥ 1 9 · 1097

≥ 5 2 · 10109 AG(3,5) [1546]

IV.2.2

662

156

806

31

6

1

≥ 1 6 · 10116

≥ 1 PG(3,5) [1546]

IV.2.2

663

63

279

31

7

3

≥ 1

≥ 1 [1042, 1]

VI.16.87

664

32

124

31

8

7

≥ 1

≥ 1 D#730* [1042, 1714]

VI.16.87

665

63

217

31

9

4

≥ 1

≥ 1 [1]

VI.16.87

666

280

868

31

10

1

?

?

667

435

899

31

15

1

?

?

668

32

62

31

16

15

≥ 1028

≥ 1 R#681,AG4(5,2),HD [1319]

669

63

93

31

21

10

≥ 1017

0 R#680* [1900, 2058]

670

125

155

31

25

6

≥ 1012

≥ 1012 R#679,AG2(3, 5) [1223]

671

156

186

31

26

5

?

0 R#678*,×3

672

280

310

31

28

3

?

0 R#677*,×3

673

435

465

31

29

2

0

0 R#676*,×2

674

900

930

31

30

1

0

0 R#675*,×2,AG(2,30)

675

931

931

31

31

1

0

- ×1,PG(2,30)

676

466

466

31

31

2

0

-×1

677

311

311

31

31

3

?

-

678

187

187

31

31

5

0

-×1

679

156

156

31

31

6

≥ 1017

-PG2(3, 5) [1223]

VI.18.73

680

94

94

31

31

10

0

-×1

681

63

63

31

31

15

≥ 1017

-PG4(5, 2) [1223]

VI.18.73

682

33

352

32

3

2

≥ 1013

≥ 4.4 · 106 2#144,D#775* [562]

683

9

96

32

3

8

≥ 107

≥ 105 8#2 [1548]

684

97

776

32

4

1

≥ 5985

- [332, 396]

VI.16.61

685

49

392

32

4

2

≥ 770

- 2#146 [396]

686

33

264

32

4

3

≥ 1

- D#776* [1042]

II.7.46

687

25

200

32

4

4

≥ 1022

- 4#22 [1548]

688

17

136

32

4

6

≥ 1

- 2#148

689

13

104

32

4

8

≥ 108

- 8#3 [1548]

690

9

72

32

4

12

≥ 108

- 4#24 [1548]

691

65

416

32

5

2

≥ 3

≥ 1 2#151

692

81

432

32

6

2

≥ 1

- 2#152* [1042]

IV.2.7

693

33

176

32

6

5

≥ 1

- D#777* [1042]

VI.16.18

694

21

112

32

6

8

≥ 1

- 4#25*,2#153

695

49

224

32

7

4

≥ 1052

≥ 1052 4#26 [1548]

696

225

900

32

8

1

?

-

697

113

452

32

8

2

≥ 1

- 2#155* [957]

VI.16.64

698

57

228

32

8

4

≥ 1063

- 4#27 [1548]

699

33

132

32

8

7

≥ 1

- D#778 [1042]

II.7.46

700

29

116

32

8

8

≥ 3

- 4#28*,2#157

701

17

68

32

8

14

≥ 12

- 2#158

702

145

464

32

10

2

?

- 2#159*

703

25

80

32

10

12

≥ 44

- 2#160

704

33

96

32

11

10

≥ 20

? 2#161,D#779*

705

177

472

32

12

2

?

- 2#162*

706

45

120

32

12

8

≥ 1

- 2#163

707

33

88

32

12

11

≥ 1

- D#780* [1042]

708

65

160

32

13

6

?

? 2#164*

709

105

240

32

14

4

?

- 2#165*

710

225

480

32

15

2

?

? 2#166*

711

481

962

32

16

1

?

-

712

241

482

32

16

2

?

- 2#167*

713

161

322

32

16

3

?

-

714

121

242

32

16

4

≥ 1

- 2#168* [1]

VI.16.31

715

97

194

32

16

5

?

-

716

81

162

32

16

6

?

- 2#169*

717

61

122

32

16

8

≥ 1

- 2#170

718

49

98

32

16

10

≥ 45

- 2#171

719

41

82

32

16

12

≥ 115308

- 2#172

720

33

66

32

16

15

≥ 1

- D#781 [2110]

721

305

488

32

20

2

?

-

722

369

492

32

24

2

?

-

723

93

124

32

24

8

?

- R#730*

724

217

248

32

28

4

?

- R#729*

725

465

496

32

30

2

0

- R#728*,×2

726

961

992

32

31

1

≥ 1

≥ 1 R#727,AG(2,31)

727

993

993

32

32

1

≥ 1

- PG(2,31)

VI.18.82

728

497

497

32

32

2

0

-×1

729

249

249

32

32

4

?

-

730

125

125

32

32

8

0

-×1

731

67

737

33

3

1

≥ 1035

- [1042, 1463]

VI.16.12

732

34

374

33

3

2

≥ 1041

- D#811* [1548]

VI.16.81

733

23

253

33

3

3

≥ 2 · 1014

- [1548]

VI.16.24

734

12

132

33

3

6

≥ 108

≥ 1 3#55 [1548]

735

7

77

33

3

11

155118

-[1545]

736

100

825

33

4

1

≥ 5985

≥ 1 [332, 396, 1047]

IV.2.2

737

12

99

33

4

9

≥ 1010

≥ 1 3#56

738

45

297

33

5

3

≥ 1054

≥ 1 3#57 [2, 1548]

739

166

913

33

6

1

?

-

740

56

308

33

6

3

≥ 1

- [1042]

IV.2.7

741

34

187

33

6

5

≥ 1

- D#812* [1042]

VI.16.86

742

16

88

33

6

11

≥ 1

- #13+#484

743

12

66

33

6

15

≥ 11604

≥ 12 3#58

744

232

957

33

8

1

≥ 1

≥ 1 [688]

745

45

165

33

9

6

≥ 35805

? 3#59 [1224]

746

100

330

33

10

3

≥ 1

? 3#60* [528]

IV.2.67

747

331

993

33

11

1

?

-

748

166

498

33

11

2

?

-

749

111

333

33

11

3

≥ 1

- 3#61* [528]

IV.2.67

750

67

201

33

11

5

≥ 1

- [2144]

VI.16.30

751

56

168

33

11

6

≥ 1071

- 3#62 [1548]

752

34

102

33

11

10

≥ 1

- D#813* [1042]

VI.16.88

753

31

93

33

11

11

≥ 1

- [1042]

II.7.46

754

23

69

33

11

15

≥ 1103

- 3#63

755

364

1001

33

12

1

?

-

756

155

341

33

15

3

?

-

757

496

1023

33

16

1

≥ 1

≥ 1 [1877]

758

34

66

33

17

16

≥ 1

0 R#781,×3 [1241, 2110]

759

133

209

33

21

5

?

-

760

56

88

33

21

12

?

- R#780*

761

694

1041

33

22

1

?

-

762

232

348

33

22

3

?

-

763

100

150

33

22

7

?

-

764

78

117

33

22

9

?

-

765

64

96

33

22

11

?

- R#779*

766

760

1045

33

24

1

?

-

767

100

132

33

25

8

≥ 1

0 R#778,×3 [1241, 1013]

768

144

176

33

27

6

?

- R#777*

769

232

264

33

29

4

?

0 R#776*,×3

770

320

352

33

30

3

?

- R#775*

771

496

528

33

31

2

?

0 R#774*,×3

772

1024

1056

33

32

1

≥ 11

≥ 11 R#773,AG(2,32) [679]

773

1057

1057

33

33

1

≥ 6

- PG(2,32) [679]

VI.18.28

774

529

529

33

33

2

?

-

775

353

353

33

33

3

0

-×1

776

265

265

33

33

4

?

-

777

177

177

33

33

6

?

-

778

133

133

33

33

8

≥ 1

- [1013]

VI.18.73

779

97

97

33

33

11

?

-

780

89

89

33

33

12

0

-×1

781

67

67

33

33

16

≥ 1

- [2110]

VI.18.73

782

69

782

34

3

1

≥ 4 · 1041

≥ 1 [1042, 1047, 1463]

VI.16.12

783

18

204

34

3

4

≥ 4 · 1014

≥ 1 2#173

784

52

442

34

4

2

≥ 207

≥ 1 2#174

785

18

153

34

4

6

≥ 1

- [1042]

VI.16.84

786

35

238

34

5

4

≥ 2

≥ 2 2#175,D#869*

787

171

969

34

6

1

≥ 1

- [46]

II.3.32

788

18

102

34

6

10

≥ 4

≥ 3 2#176

789

35

170

34

7

6

≥ 3

≥ 1 2#177,D#870* [43]

VI.16.87

790

120

510

34

8

2

≥ 1

≥ 1 2#178

791

18

68

34

9

16

≥ 103

≥ 1 2#179

792

35

119

34

10

9

≥ 1

- D#871* [1042]

793

341

1054

34

11

1

?

?

794

52

136

34

13

8

≥ 1

? 2#180

795

35

85

34

14

13

≥ 1

- D#872* [1]

796

120

272

34

15

4

?

? 2#181*

797

256

544

34

16

2

477603

477603 2#182 [1225]

798

545

1090

34

17

1

?

-

799

273

546

34

17

2

≥ 1012

- 2#183 [1239]

800

137

274

34

17

4

≥ 1

- 2#184* [382]

801

69

138

34

17

8

≥ 1

- 2#185

802

35

70

34

17

16

≥ 1854

- 2#186,D#873

803

715

1105

34

22

1

?

-

804

69

102

34

23

11

?

0 R#813*,×3

805

154

187

34

28

6

?

- R#812*

806

341

374

34

31

3

?

0 R#811*,×3

807

528

561

34

32

2

0

- R#810*,×2

808

1089

1122

34

33

1

0

0 R#809*,×2,AG(2,33)

809

1123

1123

34

34

1

0

-×1

810

562

562

34

34

2

0

-×1

811

375

375

34

34

3

?

-

812

188

188

34

34

6

0

-×1

813

103

103

34

34

11

?

-

814

36

420

35

3

2

≥ 2 · 1050

≥ 1 D#961* [1040, 1548]

VI.16.81

815

15

175

35

3

5

≥ 1015

≥ 1011 5#14 [1548]

816

6

70

35

3

14

48

0 7#4 [1174, 1548]

817

36

315

35

4

3

≥ 1

≥ 1 D#962* [1042, 144]

VI.16.83

818

16

140

35

4

7

≥ 1015

≥ 1015 7#5 [1548]

819

8

70

35

4

15

≥ 2224

82 5#15 [697, 1271]

820

141

987

35

5

1

≥ 1

- [1042]

VI.16.16

821

71

497

35

5

2

≥ 1

- [1042]

VI.16.17

822

36

252

35

5

4

≥ 2

- D#963* [1042, 2130]

VI.16.85

823

29

203

35

5

5

≥ 2

- [1042, 380]

II.7.46

824

21

147

35

5

7

≥ 1024

- 7#6 [1548]

825

15

105

35

5

10

≥ 1

≥ 1 5#16*,#102+#280 [1]

826

11

77

35

5

14

≥ 106

- 7#7 [1548]

827

36

210

35

6

5

≥ 1

≥ 1 #103+#282,D#964* [1042, 144]

828

211

1055

35

7

1

?

-

829

106

530

35

7

2

≥ 1

- [4]

830

71

355

35

7

3

≥ 1

- [2144]

VI.16.30

831

43

215

35

7

5

≥ 1

- 5#18*,#106+#287

832

36

180

35

7

6

≥ 1

- D#965* [1042]

II.7.46

833

31

155

35

7

7

≥ 5

- PG2 (4, 2) [2043]

II.7.46

834

22

110

35

7

10

≥ 1

- 5#19*,#108+#288

835

16

80

35

7

14

≥ 1

- #131↓#259

836

15

75

35

7

15

- 5#20 [1548]

837

36

140

35

9

8

≥ 4

≥ 6 D#966* [1042, 1629]

VI.16.87

838

316

1106

35

10

1

?

-

839

106

371

35

10

3

?

-

840

64

224

35

10

5

≥ 1

- [22]

II.3.32

841

46

161

35

10

7

?

-

842

36

126

35

10

9

≥ 2

- D#967* [1042, 2130]

843

22

77

35

10

15

≥ 1

- #104↓#290

844

176

560

35

11

2

?

?

845

36

105

35

12

11

≥ 1

≥ 1 D#968* [1042, 1566]

VI.16.88

846

456

1140

35

14

1

?

-

847

92

230

35

14

5

?

-

848

66

165

35

14

7

?

-

849

36

90

35

14

13

≥ 1

- D#969* [1042]

850

246

574

35

15

2

?

-

851

99

231

35

15

5

?

-

852

36

84

35

15

14

≥ 1

- D#970*, #142+#264

853

176

385

35

16

3

?

?

854

561

1155

35

17

1

?

?

855

36

70

35

18

17

≥ 91

≥ 91 R#873,HD [404, 2110]

856

96

168

35

20

7

?

-

857

351

585

35

21

2

?

-

858

141

235

35

21

5

?

-

859

51

85

35

21

14

?

- R#872*

860

85

119

35

25

10

?

- R#871*

861

316

395

35

28

3

?

-

862

136

170

35

28

7

?

- R#870*

863

64

80

35

28

15

?

-

864

204

238

35

30

5

?

- R#869*

865

561

595

35

33

2

0

0 R#868*,x2

866

1156

1190

35

34

1

?

? R#867*,AG(2,34)

867

1191

1191

35

35

1

?

- PG(2,34)

868

596

596

35

35

2

0

-×1

869

239

239

35

35

5

?

-

870

171

171

35

35

7

?

-

871

120

120

35

35

10

?

-

872

86

86

35

35

14

0

-×1

873

71

71

35

35

17

≥ 9

- [2110, 619]

VI.18.73

874

73

876

36

3

1

≥ 1034

- [1042, 1463]

VI.16.12

875

37

444

36

3

2

≥ 1010

- 2#187,D#991*

876

25

300

36

3

3

≥ 1025

- 3#64 [1548]

877

19

228

36

3

4

≥ 1017

- 4#29 [1548]

878

13

156

36

3

6

≥ 1017

- 6#8 [1548]

879

10

120

36

3

8

≥ 1012

- 4#30 [1548]

880

9

108

36

3

9

≥ 1014

≥ 105 9#2 [1707]

881

7

84

36

3

12

412991

- 12#1 [1545]

882

109

981

36

4

1

≥ 1.6 · 1013

- [396]

VI.16.61

883

55

495

36

4

2

≥ 1

- [1042]

IV.2.7

884

37

333

36

4

3

≥ 1040

- 3#68,D#992* [1548]

885

28

252

36

4

4

≥ 1032

≥ 1026 4#32 [1548]

886

19

171

36

4

6

≥ 1020

- 3#69

887

13

117

36

4

9

≥ 108

- 9#3 [1548]

888

10

90

36

4

12

≥ 109

- 6#10 [1548]

889

145

1044

36

5

1

≥ 1

≥ 1 [1042, 41]

VI.16.70

890

25

180

36

5

6

≥ 1038

≥ 1038 6#11 [1548]

891

10

72

36

5

16

≥ 108

27121734 4#33,2#195 [1548, 1633]

892

181

1086

36

6

1

≥ 1

- [1042, 2144]

VI.16.54

893

91

546

36

6

2

≥ 5

- 2#196

894

61

366

36

6

3

≥ 1

- 3#73* [1042]

VI.16.18

895

46

276

36

6

4

≥ 1

- 4#34*,2#197

896

37

222

36

6

5

≥ 2

- D#993 [1042]

897

31

186

36

6

6

≥ 1051

- 6#12 [1548]

898

21

126

36

6

9

≥ 1

-3#75

899

19

114

36

6

10

≥ 1

- 2#199

900

16

96

36

6

12

≥ 1019

- 6#13 [1548]

901

13

78

36

6

15

≥ 1011

- 3#77 [1548]

902

217

1116

36

7

1

≥ 1

? [1537]

VI.16.30

903

28

144

36

7

8

≥ 5432

≥ 1 4#36 [1, 1224]

904

64

288

36

8

4

≥ 1077

≥ 1077 4#37 [1548]

905

22

99

36

8

12

≥ 1

- 3#78* [1042]

VI.16.30

906

289

1156

36

9

1

?

-

907

145

580

36

9

2

≥ 1

- 2#203* [959]

908

97

388

36

9

3

≥ 1

-[1]

VI.16.30

909

73

292

36

9

4

≥ 1090

- 4#38 [1548]

910

49

196

36

9

6

≥ 5

- 2#205* [1151]

911

37

148

36

9

8

≥ 1037

- 4#39,D#994* [1548]

912

33

132

36

9

9

≥ 1039

- 3#79

II.7.46

913

25

100

36

9

12

≥ 1028

- 4#40 [1548]

914

19

76

36

9

16

≥ 1016

- 4#41 [1548]

915

325

1170

36

10

1

?

-

916

55

198

36

10

6

?

- 3#80*,2#209*

917

121

396

36

11

3

≥ 10188

≥ 10188 3#81 [1548]

918

397

1191

36

12

1

?

-

919

199

597

36

12

2

?

-

920

133

399

36

12

3

≥ 10208

- 3#82 [1548]

921

100

300

36

12

4

?

- 2#210*

922

67

201

36

12

6

≥ 1

- 3#83* [2144]

VI.16.30

923

45

135

36

12

9

≥ 1057

- 3#84

924

37

111

36

12

11

≥ 1

- D#995* [1042]

II.7.46

925

34

102

36

12

12

≥ 2

- 3#85*,2#211

VI.16.88

926

469

1206

36

14

1

?

-

927

505

1212

36

15

1

?

-

928

85

204

36

15

6

?

- 2#212*

929

136

306

36

16

4

?

- 2#213*

930

289

612

36

17

2

≥ 3481

≥ 3481 2#214 [1225]

931

613

1226

36

18

1

?

-

932

307

614

36

18

2

≥ 8 · 1013

- 2#215 [1239]

933

205

410

36

18

3

?

-

934

154

308

36

18

4

?

- 2#216*

935

103

206

36

18

6

?

- 2#217*

936

69

138

36

18

9

?

-

937

52

104

36

18

12

?

- 2#218*

938

37

74

36

18

17

≥ 1

- D#996 [1042]

939

685

1233

36

20

1

?

-

940

115

207

36

20

6

?

-

941

721

1236

36

21

1

?

-

942

91

156

36

21

8

?

-

943

49

84

36

21

15

?

- R#970*

944

253

414

36

22

3

?

-

945

55

90

36

22

14

?

- R#969*

946

208

312

36

24

4

?

-

947

70

105

36

24

12

?

- R#968*

948

91

126

36

26

10

?

- R#967*

949

105

140

36

27

9

?

- R#966*

950

973

1251

36

28

1

?

-

951

145

180

36

29

7

?

0 R#965*,×3

952

1045

1254

36

30

1

?

-

953

175

210

36

30

6

?

- R#964*

954

217

252

36

31

5

?

0 R#963*,×3

955

280

315

36

32

4

?

- R#962*

956

385

420

36

33

3

?

- R#961*

957

595

630

36

34

2

?

- R#960*

958

1225

1260

36

35

1

?

? R#959*,AG(2,35)

959

1261

1261

36

36

1

?

- PG(2,35)

960

631

631

36

36

2

?

-

961

421

421

36

36

3

?

-

962

316

316

36

36

4

0

-×1

963

253

253

36

36

5

?

-

964

211

211

36

36

6

0

-×1

965

181

181

36

36

7

?

-

966

141

141

36

36

9

?

-

967

127

127

36

36

10

?

-

968

106

106

36

36

12

0

-×1

969

91

91

36

36

14

0

-×1

970

85

85

36

36

15

?

-

971

75

925

37

3

1

≥ 10196

≥ 1 [1013, 1042, 1174]

VI.16.12

972

112

1036

37

4

1

≥ 1.69 · 1013

≥ 210 [1042, 396]

IV.2.2

973

75

555

37

5

2

≥ 1

≥ 1 [1042, 41]

VI.16.31

974

186

1147

37

6

1

≥ 1

≥ 1 [1042, 958]

IV.2.2

975

112

592

37

7

2

≥ 1

? [1042]

976

297

1221

37

9

1

?

?

977

408

1258

37

12

1

?

?

978

75

185

37

15

7

≥ 1

≥ 1 [1714]

979

112

259

37

16

5

?

?

980

630

1295

37

18

1

?

?

981

38

74

37

19

18

≥ 1

0 R#996,×3 [1241, 2110]

982

75

111

37

25

12

?

0 R#995*,×3

983

112

148

37

28

9

?

? R#994*

984

186

222

37

31

6

≥ 1

0 R#993,×3 [1241, 1773]

985

297

333

37

33

4

?

0 R#992*,×3

986

408

444

37

34

3

?

0 R#991*,×3

987

630

666

37

35

2

0

0 R#990*,×2

988

1296

1332

37

36

1

?

? R#989*,AG(2,36)

989

1333

1333

37

37

1

?

- PG(2,36)

990

667

667

37

37

2

0

-×1

991

445

445

37

37

3

0

-×1

992

334

334

37

37

4

0

-×1

993

223

223

37

37

6

≥ 1

- [1773]

994

149

149

37

37

9

?

-

995

112

112

37

37

12

?

-

996

75

75

37

37

18

≥ 1

- [1839]

V.1.39

997

39

494

38

3

2

≥ 1044

≥ 89 2#219,D#1071*

998

58

551

38

4

2

≥ 1

- [1042]

IV.2.7

999

20

190

38

4

6

≥ 1

≥ 4 2#220

1000

20

152

38

5

8

≥ 1

≥ 3 2#221

1001

96

608

38

6

2

≥ 1

≥ 1 2#222 [958]

1002

39

247

38

6

5

≥ 1

- D#1072* [1042]

VI.16.18

1003

77

418

38

7

3

≥ 2

? [1042, 1]

VI.16.70

1004

20

95

38

8

14

≥ 1

- [1042]

VI.16.88

1005

153

646

38

9

2

≥ 1

? 2#223* [958, 1548]

VI.16.31

1006

115

437

38

10

3

?

-

1007

20

76

38

10

18

≥ 1016

≥ 4 2#224

1008

77

266

38

11

5

?

?

1009

210

665

38

12

2

?

-

1010

39

114

38

13

12

≥ 1

? 2#225*,D#1073* [1042]

VI.16.88

1011

96

228

38

16

6

?

? 2#226*

1012

153

342

38

17

4

?

? 2#227*

1013

324

684

38

18

2

≥ 1

? 2#228* [528]

IV.2.67

1014

685

1370

38

19

1

?

-

1015

343

686

38

19

2

?

- 2#229*

1016

229

458

38

19

3

?

-

1017

172

344

38

19

4

?

- 2#230*

1018

115

230

38

19

6

?

- 2#231*

1019

77

154

38

19

9

?

-

1020

58

116

38

19

12

?

- 2#232*

1021

39

78

38

19

18

≥ 5 87 · 1014

- 2#233,D#1074 [1374]

1022

666

703

38

36

2

?

- R#1025*

1023

1369

1406

38

37

1

≥ 1

≥ 1 R#1024,AG(2,37)

1024

1407

1407

38

38

1

≥ 1

- PG(2,37)

VI.18.82

1025

704

704

38

38

2

?

-

1026

79

1027

39

3

1

≥ 1056

- [1463]

VI.16.12

1027

40

520

39

3

2

≥ 6 · 1024

- D#1163* [1548]

VI.16.13

1028

27

351

39

3

3

≥ 1028

≥ 1030 3#86 [1548]

1029

14

182

39

3

6

≥ 2 · 1034

- [1548]

VI.16.82

1030

7

91

39

3

13

1033129

- 13#1 [696]

1031

40

390

39

4

3

≥ 1033

≥ 1033 3#87,D#1164* [1548]

1032

40

312

39

5

4

≥ 1

≥ 1 D#1165* [1, 1042]

VI.16.85

1033

196

1274

39

6

1

≥ 1

- [18]

II.3.32

1034

66

429

39

6

3

≥ 1

≥ 1 3#88 [1833]

1035

40

260

39

6

5

≥ 1

- D#1166* [1042]

VI.16.86

1036

16

104

39

6

13

≥ 1

- #13+#742

1037

14

91

39

6

15

≥ 1

- [1042]

VI.16.86

1038

14

78

39

7

18

≥ 1011

0 3#89 [1548]

1039

40

195

39

8

7

≥ 1

≥ 1 D#1167* [1, 1042]

VI.16.87

1040

105

455

39

9

3

≥ 1

- [21]

II.3.32

1041

27

117

39

9

12

≥ 1017

≥ 1017 3#90 [1548]

1042

40

156

39

10

9

≥ 1

≥ 3 3#91*,D#1168* [1042, 1629]

VI.16.87

1043

66

234

39

11

6

≥ 21584

? 3#92 [1224]

1044

144

468

39

12

3

≥ 1

? 3#93* [528]

IV.2.67

1045

40

130

39

12

11

≥ 1

- D#1169* [1042]

VI.16.88

1046

469

1407

39

13

1

?

-

1047

235

705

39

13

2

?

-

1048

157

471

39

13

3

≥ 1

- 3#94* [528]

IV.2.67

1049

118

354

39

13

4

?

-

1050

79

237

39

13

6

≥ 10125

- 3#95 [1548]

1051

53

159

39

13

9

≥ 1

- 3#96* [1]

1052

40

120

39

13

12

≥ 1033

- 3#97,D#1170 [1548]

1053

37

111

39

13

13

≥ 1

- [1042]

II.7.46

1054

27

81

39

13

18

≥ 208311

- 3#98

1055

40

104

39

15

14

≥ 1

- D#1171* [1042]

1056

222

481

39

18

3

?

-

1057

703

1443

39

19

1

?

?

1058

40

78

39

20

19

≥ 1

≥ 1 R#1074,HD [2110]

1059

196

364

39

21

4

?

-

1060

976

1464

39

26

1

?

-

1061

326

489

39

26

3

?

-

1062

196

294

39

26

5

?

-

1063

76

114

39

26

13

?

- R#1073*

1064

66

99

39

26

15

?

-

1065

209

247

39

33

6

?

- R#1072*

1066

456

494

39

36

3

?

- R#1071*

1067

703

741

39

37

2

0

0 R#1070*,×2

1068

1444

1482

39

38

1

0

0 R#1069*,×2,AG(2,38)

1069

1483

1483

39

39

1

0

- ×1, PG(2,38)

1070

742

742

39

39

2

0

-×1

1071

495

495

39

39

3

?

-

1072

248

248

39

39

6

0

-×1

1073

115

115

39

39

13

0

-×1

1074

79

79

39

39

19

≥ 2091

- [1087]

VI.18.73

1075

81

1080

40

3

1

≥ 1048

≥ 107 AG(4,3) [1463, 1548]

VI.16.12

1076

21

280

40

3

4

≥ 1024

≥ 1021 4#42 [1548]

1077

9

120

40

3

10

≥ 108

≥ 105 10#2 [1548]

1078

6

80

40

3

16

76

1 8#4,4#43 [1548]

1079

121

1210

40

4

1

≥ 1013

- [332, 375]

VI.16.61

1080

61

610

40

4

2

≥ 104

- 2#237

1081

41

410

40

4

3

≥ 1

- D#1192*

II.7.46

1082

31

310

40

4

4

≥ 1

- 2#238

1083

25

250

40

4

5

≥ 1023

- 5#22 [1548]

1084

21

210

40

4

6

≥ 1

- 2#239

1085

16

160

40

4

8

≥ 1015

≥ 1015 8#5 [1548]

1086

13

130

40

4

10

≥ 1014

- 10#3 [1548]

1087

11

110

40

4

12

≥ 1

- 2#242

1088

9

90

40

4

15

≥ 106

- 5#24 [1548]

1089

161

1288

40

5

1

≥ 1

- [1042]

VI.16.16

1090

81

648

40

5

2

≥ 1

- 2#243

1091

41

328

40

5

4

≥ 1046

- 4#45,D#1193*

1092

33

264

40

5

5

≥ 1

- [1042]

VI.16.17

1093

21

168

40

5

8

≥ 1024

- 8#6 [1548]

1094

17

136

40

5

10

≥ 1

- 2#246

1095

11

88

40

5

16

≥ 107

- 8#7 [1548]

1096

201

1340

40

6

1

≥ 1

-[18]

II.3.32

1097

51

340

40

6

4

≥ 1

- 4#48*,2#248

1098

21

140

40

6

10

≥ 1

- 5#25*,2#249

1099

49

280

40

7

5

≥ 1053

≥ 1053 5#26 [1548]

1100

21

120

40

7

12

≥ 1018

≥ 1 4#49,2#250

1101

281

1405

40

8

1

?

-

1102

141

705

40

8

2

≥ 1

- [19]

1103

71

355

40

8

4

≥ 1

- [2144]

VI.16.30

1104

57

285

40

8

5

≥ 1063

- 5#27 [1548]

1105

41

205

40

8

7

≥ 1

- D#1194* [1042]

II.7.46

1106

36

180

40

8

8

≥ 3

- 4#50*,2#251

1107

29

145

40

8

10

≥ 1

- 5#28*,#157+#379

1108

21

105

40

8

14

≥ 1

- [1042]

VI.16.30

1109

81

360

40

9

4

≥ 10108

≥ 10108 4#51 [1548]

1110

361

1444

40

10

1

?

-

1111

181

724

40

10

2

≥ 2

- 2#253* [1, 959]

VI.16.30

1112

121

484

40

10

3

≥ 1

- [1]

VI.16.31

1113

91

364

40

10

4

≥ 10125

- 4#52 [1548]

1114

73

292

40

10

5

≥ 1

- [2144]

VI.16.30

1115

61

244

40

10

6

≥ 10

- 2#255* [272]

1116

46

184

40

10

8

≥ 1

- 4#53*,2#256* [1]

1117

41

164

40

10

9

≥ 1

- D#1195* [1042]

II.7.46

1118

37

148

40

10

10

≥ 1

- 2#257

1119

31

124

40

10

12

≥ 152

- 4#54

1120

25

100

40

10

15

≥ 1

- #160+#384

1121

21

84

40

10

18

≥ 5

- 2#259

1122

441

1470

40

12

1

?

-

1123

111

370

40

12

4

?

- 2#260*

1124

45

150

40

12

10

≥ 1

- 2#261*,#84+#543

1125

481

1480

40

13

1

?

?

1126

105

300

40

14

5

?

-

1127

561

1496

40

15

1

?

-

1128

141

376

40

15

4

?

- 2#262*

1129

81

216

40

15

7

≥ 1

- [2144]

1130

57

152

40

15

10

?

- 2#263*

1131

36

96

40

15

16

≥ 1

- 2#264

1132

76

190

40

16

8

≥ 1

- 2#265

1133

171

380

40

18

4

?

- 2#266*

1134

361

760

40

19

2

≥ 7417

≥ 7417 2#267 [1225]

1135

761

1522

40

20

1

?

-

1136

381

762

40

20

2

≥ 3 · 1016

- 2#268 [1239]

1137

191

382

40

20

4

?

- 2#269*

1138

153

306

40

20

5

?

-

1139

96

192

40

20

8

≥ 3

- 2#270

1140

77

154

40

20

10

?

- 2#271*

1141

41

82

40

20

19

≥ 1

- D#1196 [2110]

1142

121

220

40

22

7

?

-

1143

921

1535

40

24

1

?

-

1144

231

385

40

24

4

?

-

1145

93

155

40

24

10

?

-

1146

65

104

40

25

15

?

- R#1171*

1147

1001

1540

40

26

1

?

-

1148

81

120

40

27

13

≥ 1013

≥ 1013 R#1170,AG3 (4, 3) [1223]

1149

217

310

40

28

5

?

-

1150

91

130

40

28

12

?

- R#1169*

1151

1161

1548

40

30

1

?

-

1152

291

388

40

30

4

?

-

1153

117

156

40

30

10

?

- R#1168*

1154

156

195

40

32

8

?

- R#1167*

1155

221

260

40

34

6

?

- R#1166*

1156

273

312

40

35

5

?

- R#1165*

1157

351

390

40

36

4

?

- R#1164*

1158

481

520

40

37

3

?

0 R#1163*,×3

1159

741

780

40

38

2

0

- R#1162*,×2

1160

1521

1560

40

39

1

?

? R#1161*,AG(2,39)

1161

1561

1561

40

40

1

?

- PG(2,39)

1162

781

781

40

40

2

0

-×1

1163

521

521

40

40

3

?

-

1164

391

391

40

40

4

?

-

1165

313

313

40

40

5

0

-×1

1166

261

261

40

40

6

0

-×1

1167

196

196

40

40

8

0

-×1

1168

157

157

40

40

10

0

-×1

1169

131

131

40

40

12

?

-

1170

121

121

40

40

13

≥ 1029

-PG3(4,3) [1223]

VI.18.73

1171

105

105

40

40

15

≥ 4

- [1183]

1172

42

574

41

3

2

≥ 6 · 1024

≥ 1 D [1040, 1548]

VI.16.81

1173

124

1271

41

4

1

≥ 2

≥ 1 [332, 1042, 1047]

III.2.9

1174

165

1353

41

5

1

≥ 15

≥ 1 [396, 1042]

III.2.9

1175

42

287

41

6

5

≥ 2

≥ 2 D [1, 2, 1042]

VI.16.86

1176

42

246

41

7

6

≥ 1

≥ 1 D [1, 1042]

VI.16.87

1177

288

1476

41

8

1

≥ 1

≥ 1 [958]

II.7.50

1178

370

1517

41

10

1

?

?

1179

247

779

41

13

2

?

?

1180

42

123

41

14

13

≥ 1

?D[1]

1181

247

533

41

19

3

?

?

1182

780

1599

41

20

1

?

?

1183

42

82

41

21

20

≥ 1

0 R#1196,×3 [2110]

1184

124

164

41

31

10

?

0 R#1195*,×3

1185

165

205

41

33

8

?

0 R#1194*,×3

1186

288

328

41

36

5

?

0 R#1193*,×3

1187

370

410

41

37

4

?

0 R#1192*,×3

1188

780

820

41

39

2

?

0 R#1191*,×3

1189

1600

1640

41

40

1

?

? R#1190*,AG(2,40)

1190

1641

1641

41

41

1

?

- PG(2,40)

1191

821

821

41

41

2

?

-

1192

411

411

41

41

4

?

-

1193

329

329

41

41

5

?

-

1194

206

206

41

41

8

0

-×1

1195

165

165

41

41

10

?

-

1196

83

83

41

41

20

≥ 1

- [2110]

VI.18.73

 BIBDs with block size 3. Existence results on designs with “small” block size. Resolvable designs. Symmetric designs. Steiner systems. PBD constructions make BIBDs. [1016] A general introduction to combinatorics and in particular to design theory. [1271] Contains exhaustive catalogues of BIBDs and their resolutions in electronic form.

References Cited: [1, 2, 4, 18, 19, 21, 22, 33, 41, 43, 46, 114, 124, 127, 144, 180, 246, 248, 272, 323, 324, 331, 332, 352, 375, 380, 382, 392, 393, 396, 404, 416, 514, 518, 528, 534, 537, 562, 587, 607, 619, 620, 623, 660, 679, 688, 692, 693, 694, 696, 697, 710, 734, 790, 827, 848, 856, 898, 957, 958, 959, 976, 978, 1005, 1013, 1016, 1034, 1040, 1042, 1044, 1047, 1053, 1087, 1138, 1151, 1172, 1174, 1183, 1184, 1185, 1188, 1207, 1208, 1223, 1224, 1225, 1228, 1236, 1239, 1241, 1243, 1251, 1262, 1263, 1264, 1265, 1267, 1268, 1271, 1319, 1321, 1339, 1346, 1347, 1349, 1372, 1373, 1374, 1375, 1377, 1378, 1379, 1380, 1381, 1398, 1463, 1496, 1497, 1533, 1537, 1543, 1544, 1545, 1546, 1547, 1548, 1549, 1551, 1556, 1566, 1578, 1588, 1605, 1606, 1607, 1609, 1610, 1612, 1628, 1629, 1632, 1633, 1665, 1704, 1705, 1707, 1714, 1720, 1729, 1743, 1773, 1832, 1833, 1839, 1877, 1900, 1903, 1912, 1938, 1940, 1941, 1943, 1944, 1945, 1946, 1976, 1999, 2031, 2039, 2043, 2045, 2058, 2059, 2060, 2062, 2063, 2064, 2065, 2066, 2067, 2071, 2090, 2110, 2130, 2144, 2148]

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