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

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.

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2-( v , k , λ) Designs of Small Order

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 X ⊆ V and $D \subseteq B$ . The subdesign is proper if X ⊂ V.

1.4 Proposition If a (v, k, λ) design has a proper (w, k, λ) subdesign, then $w \leq \frac{v - 1}{k - 1}$ .

1.5 Remark The three parameters v, k, and λ determine the remaining two as $r = \frac{λ (v - 1)}{k - 1}$ and $b = \frac{v r}{k}$ . 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 $(\begin{matrix} v \\ k \end{matrix})$ 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 $D \in D$ , the collection of blocks ${D^{'} \cap D : D^{'} \in D, D \neq D^{'}} = B$ .
Hadamard	if v = 4n − 1, k = 2n − 1, and λ = n − 1 for some integer n ≥ 2. See §V.1.
m-multiple	if $v, \frac{b}{m}, \frac{r}{m}, k, \frac{λ}{m}$ 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 $D \in D$ , the collection of blocks ${D^{'} \ D : D^{'} \in D, D \neq D^{'}} = B$ .
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 $J A = k \hat{J}$ , where I is a v × v identity matrix, J is a v × v all ones matrix, and $\hat{J}$ 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 b ≥ v.

1.10 Proposition An additional trivial necessary condition for the existence of an RBIBD is (3) k|v. A nontrivial condition is that b ≥ v + 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 ₁ → V 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 $(V, \bar{B})$ where $\bar{B} = {V \ B : B \in B}$ .

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

1.16 Remarks

In view of Proposition 1.15, one usually considers designs with v ≥ 2k and obtains the rest by taking the complement.
Complement has also been used, when $(V, B)$ is a simple 2-(v, b, r, k, λ) design, to denote the $2 - (v, (\begin{matrix} v \\ k \end{matrix}) - b, (\begin{matrix} v - 1 \\ k - 2 \end{matrix}) - r, k, (\begin{matrix} v - 2 \\ k - 2 \end{matrix}) - λ)$ 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:	00000001111112222223333444455556666	2:	00000001111112222223333444455556666
	13579bd3478bc3478bc789a789a789a789a		13579bd3478bc3478bc789a789a789a789a
	2468ace569ade65a9edbcdecbeddebcedcb		2468ace569ade65a9edbcdecbededcbdebc
3:	00000001111112222223333444455556666	4:	00000001111112222223333444455556666
	13579bd3478bc3478bc789a789a789a789a		13579bd3478bc3478bc789a789a789a789a
	2468ace569ade65a9edbcdedebcedcbcbed		2468ace569ade65a9edbcdedbecedcbcebd
5:	00000001111112222223333444455556666	6:	00000001111112222223333444455556666
	13579bd3478bc3478bc789a789a789a789a		13579bd3478bc3478bc789a789a789a789a
	2468ace569ade65a9edbceddecbcbdeedbc		2468ace569ade65a9edbceddecbcdbeebdc
7:	00000001111112222223333444455556666	8:	00000001111112222223333444455556666
	13579bd3478bc3478bc789a789a789a789a		13579bd3478bc34789c78ab789a789a789a
	2468ace569ade65a9edbdececbdcedbdbce		2468ace569ade65abedc9deedcbdebcbcde
9:	00000001111112222223333444455556666	10:	00000001111112222223333444455556666
	13579bd3478bc34789c78ab789a789a789a		13579bd3478bc34789c78ab789a789a789a
	2468ace569ade65abedc9deedbcdecbbcde		2468ace569ade65abedc9deedcbdcbebedc
11:	00000001111112222223333444455556666	12:	00000001111112222223333444455556666
	13579bd3478bc34789c789a789a78ab789a		13579bd3478bc34789c789a789a789a78ab
	2468ace569ade65abedcdbeecdbd9cebecd		2468ace569ade65abedcdbeecdbbecdd9ce
13:	00000001111112222223333444455556666	14:	00000001111112222223333444455556666
	13579bd3478bc34789c789a789a78ab789a		13579bd3478bc34789c789a78ab789a789a
	2468ace569ade65abedcebdedcbd9cebcde		2468ace569ade65abedcebdd9ceedcbbcde
15:	00000001111112222223333444455556666	16:	00000001111112222223333444455556666
	13579bd3478bc34789a78ac789a789b789a		13579bd3478bc34789a789a78bc789a789a
	2468ace569ade65bcdee9bddbecadcecebd		2468ace569ade65bcdeedcba9eddebccbed
17:	00000001111112222223333444455556666	18:	00000001111112222223333444455556666
	13579bd3478bc34789a789a789c78ab789a		13579bd3478bc34789a78ac789b789a789a
	2468ace569ade65bcdeedcbabedd9cecebd		2468ace569ade65bcede9bdadcedebccbde
19:	00000001111112222223333444455556666	20:	00000001111112222223333444455556666
	13579bd3478bc34789a789a78ab789c789a		13579bd3478bc34789a78ac789b789a789a
	2468ace569ade65bdceebdcc9deaebddceb		2468ace569ade65bdece9bdacdedbcecebd
21:	00000001111112222223333444455556666	22:	00000001111112222223333444455556666
	13579bd3478ac34789b789a789a78ab789c		13579bd3478ac34789b789a789c789a78ab
	2468ace569bde65acedbdcecedbd9ceeabd		2468ace569bde65acedbdceeabdcedbd9ce
23:	00000001111112222223333444455566667	24:	00000001111112222223333444455566667
	13579bd3478bc34589c789a58ab789789aa		13579bd3478bc34589c789a589a78978aba
	2468ace569ade67abedcdbed9cebececdbd		2468ace569ade67abedcdbebecdecdd9ceb
25:	00000001111112222223333444455566667	26:	00000001111112222223333444455566667
	13579bd3478bc34589c789a589a78978aba		13579bd3478bc3458ac789a589a789789ab
	2468ace569ade67abedcedbecbdbdcd9cee		2468ace569ade67b9edcbededbcacddecbe
27:	00000001111112222223333444455566667	28:	00000001111112222223333444455566667
	13579bd3478bc34589a78ab589c789789aa		13579bd3478bc34589a78ab589c789789aa
	2468ace569ade67bcded9ceaebdcdeebcdb		2468ace569ade67bcedd9ceaebdedccbdeb
29:	00000001111112222223333444455566667	30:	00000001111112222223333444455566667
	13579bd3478bc34589a789a58bc789789aa		13579bd3478bc34589a78ab589c789789aa
	2468ace569ade67bcedebdca9eddeccdbeb		2468ace569ade67bdced9ceabedcedecbdb
31:	00000001111112222223333444455566667	32:	00000001111112222223333444455566667
	13579bd3478bc34589a789a58bc789789aa		13579bd3478bc34589a78ab589c789789aa
	2468ace569ade67bdcedbeca9edcedecbdb		2468ace569ade67bdcec9deaebddceebdcb
33:	00000001111112222223333444455566667	34:	00000001111112222223333444455566667
	13579bd3478bc34589a78ab589c789789aa		13579bd3478bc34589a789a589c78978baa
	2468ace569ade67bdecc9deaebdecddbceb		2468ace569ade67bdecdbceaebdecdc9edb
35:	00000001111112222223333444455566678	36:	00000001111112222223333444455566678
	13579bd3478bc34569a689a579a78a789bc		13579bd3478bc34569a689a579a78a789bc
	2468ace569ade78bcdedbecedcbc9daebed		2468ace569ade78bceddbceecdbd9caebed
37:	00000001111112222223333444455566678	38:	00000001111112222223333444455566678
	13579bd3478bc34569a68ab579c78978aa9		13579bd3478bc34569a68ab579a78978ac9
	2468ace569ade78bced9dceaebdceddbebc		2468ace569ade78bdce9cdecedbadebecdb
39:	00000001111112222223333444455566678	40:	00000001111112222223333444455566678
	13579bd3478bc34569a68ab579c78978aa9		13579bd3478bc34569a689a579a78a789bc
	2468ace569ade78bdce9dceabedecdcebdb		2468ace569ade78bdcecbedecdbd9caebed
41:	00000001111112222223333444455566678	42:	00000001111112222223333444455566678
	13579bd3478bc34569a689c57ab78a789a9		13579bd3478bc34569a68ab579c78978aa9
	2468ace569ade78bdceabed9dceecdcebbd		2468ace569ade78bdec9cdeaebddeccbebd
43:	00000001111112222223333444455566678	44:	00000001111112222223333444455566678
	13579bd3478bc34569a68ac579b78978aa9		13579bd3478bc34569a689a579a78a789bc
	2468ace569ade78bdec9ebdacdeedcbcedb		2468ace569ade78bdeccbdeedcbc9daebed
45:	00000001111112222223333444455566678	46:	00000001111112222223333444455566678
	13579bd3478bc34569a68ac579a78978ab9		13579bd3478bc34569a68ab579c78978aa9
	2468ace569ade78becd9bededbcacdcdbee		2468ace569ade78becd9dceabedceddcbeb
47:	00000001111112222223333444455566678	48:	00000001111112222223333444455566678
	13579bd3478bc34569a68ab579c78978aa9		13579bd3478bc34568a69ab578c78979aa9
	2468ace569ade78bedc9cdeabeddeccdbeb		2468ace569ade79becd8dceabedcdedcbeb
49:	00000001111112222223333444455566679	50:	00000001111112222223333444455566678
	13579bd3478bc34568a689a578a78978abc		13579bd3478bc34568a69ac578a78979ab9
	2468ace569ade79becddebccdbeadec9bed		2468ace569ade79becd8bededbcadccdbee
51:	00000001111112222223333444455566678	52:	00000001111112222223333444455566678
	13579bd3478bc34568968ab579c79a78aa9		13579bd3478bc345689689a579c79a78aab
	2468ace569ade7abecd9dce8ebddcecbdbe		2468ace569ade7abecdcdeb8ebddceb9dce
53:	00000001111112222223333444455566678	54:	00000001111112222223333444455566679
	13579bd3478bc345689689a579c79a78aab		13579bd3478bc345689689a578c78a78aab
	2468ace569ade7abecdbdec8ebddcec9dbe		2468ace569ade7abecdbecd9ebdcded9cbe
55:	00000001111112222223333444455566678	56:	00000001111112222223333444455566678
	13579bd3478bc34568969ab578c78a79aa9		13579bd3478bc345689689a579c79a78aab
	2468ace569ade7abecd8cde9ebdcdedbcbe		2468ace569ade7abedcdceb8ebdcdeb9cde
57:	00000001111112222223333444455566678	58:	00000001111112222223333444455566678
	13579bd3478bc34568968ab579c79a78aa9		13579bd3478bc345689689a579c79a78aab
	2468ace569ade7abedc9cde8ebdcdedbcbe		2468ace569ade7abedcbced8ebdcded9cbe
59:	0000000111111222222333344445556667a	60:	0000000111111222222333344445556667a
	13579bd3478bc345689689a578a789789cb		13579bd3478bc34568968a9578a789789cb
	2468ace569ade7ebacdcbeddb9caecedbde		2468ace569ade7ebacddecbcb9dadeebcde
61:	00000001111112222223333444455556666	62:	00000001111112222223333444455566667
	13579bd3478ac34789b789c789a78ab789a		13579bd3478ac34589b78ab589a789789ca
	2468ace569bde65aecdbaedecdbd9cecdbe		2468ace569bde67adcec9edbedcdceeabdb
63:	00000001111112222223333444455566667	64:	00000001111112222223333444455566667
	13579bd3478ac34589b78ab589a789789ca		13579bd3478ac34589a789a589b78b789ca
	2468ace569bde67aecdd9cebcdecdeeabdb		2468ace569bde67cdbeecdbaecdd9ebaedc
65:	00000001111112222223333444455566678	66:	00000001111112222223333444455566678
	13579bd3478ac34569b68ac579a789789ba		13579bd3478ac34568968ac579a79b789ba
	2468ace569bde78dacee9bdbedcacecdbde		2468ace569bde7bdacee9bd8edcacecdbde
67:	00000001111112222223333444455566679	68:	00000001111112222223333444455566678
	13579bd3478ac34568a68ac578b789789ab		13579bd3478ac34569b68ac579a789789ba
	2468ace569bde79cbdee9bdaecdbeddacce		2468ace569bde78adcee9bdbedccdeacbde
69:	00000001111112222223333444455566679	70:	0000000111111222222333344445556667a
	13579bd3478ac3456a868ac578b789789ab		13579bd3478ac34568968ab5789789789cb
	2468ace569bde79bcede9bdaecddecbadce		2468ace569bde7dbaece9cdceabadebcdde
71:	0000000111111222222333344445556667a	72:	0000000111111222222333344445556667a
	13579bd3478ac34568968ab5789789789cb		13579bd3478ac34568968ab5789789789cb
	2468ace569bde7cabdee9cddeabbecacdde		2468ace569bde7dcbaee9cdaecbbedadcde
73:	00000001111112222223333444455566679	74:	0000000111111222222333344445556667a
	13579bd3478ac34568a689a578b78c789ab		13579bd3478ac345698689a578b789789cb
	2468ace569bde7c9bdeecdbae9dbeddacce		2468ace569bde7abdecedbcce9daedbacde
75:	0000000111111222222333344445556667a	76:	00000001111112222223333444455566678
	13579bd3478ac345689689a578b789789cb		13579bd3478ac34569b68ab578978979aac
	2468ace569bde7cdbaeedbcae9dbecacdde		2468ace569bde7da8cee9cdcbaedebcdbed
77:	00000001111112222223333444455566678	78:	00000001111112222223333444455566678
	13579bd3478ac34569a68ab578979b789ac		13579bd3478ac34569b689c578b78a79aa9
	2468ace569bde7d8cebe9cdacebdcebaded		2468ace569bde7ac8edeabd9cdedebbdcec
79:	00000001111112222223333444455566678	80:	00000001111112222223333444455566678
	13579bd3478ac34568b689c57ab79a789a9		13579bd3469ac34578b678a58ab78979c9a
	2468ace569bde79ecadaebd8dcecdbbdeec		2468ace578bde96aecdbcded9cebecaeddb

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.

00000111122223333456

147ad456945684567897

258be7b8cc79a98abbac

369cfadefefbddcfefed

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:	00000000111111122222223333344445555666778899aabbil
	134567ce34578cd34568de468bh679f78ag79b9aabcddecejm
	298dfbhkea6g9kf7c9afkg5cgfihdgifchi8ejjcjdfhgfghkn
	iaolgmjnmbohnljonblhmjjdlknmeklnekmkinlimimonooloo
2:	0000000011111112222222333334444555566667778889abil
	13457bce34589cd3456ade489eh6acf7bdg79ab9ab9abdecjm
	2689gdfka76fekg798fckh5cfgidghichfi8chjjdfgjehfgkn
	iloahnjmbmohlnjobngmljjdknmeklnekmlkinmnilmliooooo
3:	0000000011111112222222333334444555566667778889abil
	13457bde34589ce3456acd489eh6acf7bdg79ab9ab9abcdejm
	2689gcfka76fdkg798fehk5cgfidhgicfhi8hcjjfdejgfghkn
	iloahmjnbmohnljobngljmjdkmneknleklmkminlniimlooooo
4:	0000000011111112222222333334444555566667778889abil
	13457aef3458bcf34569dg489dh69ef7acg79bc9acabdcdejm
	2689bcjha769djg7b8aejh5cbfkdagkebhk8gieihdifefghkn
	ilodgknmemohklnocnfkmljgminhnilflimkjnmljnmjlooooo
5:	00000000111111122222223333344445555666778899abcfil
	134567cd34578de34568ce468ag67bh789f79aab9bacdedgjm
	298abhgkba69fhk79bgakf5cdfiedgicehi8djejjcbfghehkn
	ieolfmjnmcognjlondlhmjjhlknmfklngkmkinilmiolmnoooo
6:	000000001111111222222333334444555566667778899abcde
	123468cj23479af3458bg456ah57bi78bd89ce9adabacghiff
	5ad97fek6be8gdl7ch9em89fcn6gdkfickgjdlhmeekblhijjg
	oignbhmlkjhcinmlfjdonmeikoajlonlgmomhnkoijnfolmnok
7:	000000001111111222222333333444445555566666777889al
	124789bh2589ace39abde47abcf8bcde79cdf78adg89b9cabm
	365egifj46fhjgi5gikhf6ihjegjikfggkjehfhekiadcbdcdn
	dcaolmnk7bolmnk8olmnj9nolmknolmhmnolilmnojkjheifgo
8:	000000001111111222222333333444455566667777889aacee
	12459bdf24569cg3458bh4568bi59bh9ac78ab89cgaddbedfl
	3786cihjd78haek96kfgja7fgclgkdiimfi9djebkjcjffhmgn
	oanegklmifbmljnecolmnjdkhnmlmeojnhnoglmhloiknokoio
9:	000000001111111222222233333444455556677889abcdefgh
	13456789345678a34567894679c67ad68be9aab9bdecijkijk
	2fbcdgiadg9jcfbaehficb5b8eg89ch7adfchdfgefghmlllmn
	onklehjmmlikehnjnmgkdloilhkmjfinkgjolomnokijnnmooo
10:	000000001111111222222233333444455566577889abccdfgh
	13456789345678a345678b467ad679e69c9b89aabfghdeeijk
	2jadebchekbdc9f9ciaedg5f8bg8gbf7afcihjdiemllikjlmn
	onmkgfliilnmhgjljmhnfkokmchnidhlegojjkokonnmnmlooo
11:	0000000011111112222222333334444555566778899acdefgh
	1345678b3456789345678a467bc679d68ae9daebcabbkijijk
	2iace9fgdjbgcah9cibhdf5a8fg8bfh79gfcfdgehedclmllmn
	ojmdhnklekniflmlekmjgnokmhnniglljhmojokoiikjmnnooo
12:	0000000011111112222222333334444555566778899accdfgh
	1345678a345678b34567894679e67ac68bd9fagbfabbdeeijk
	2j9ebcmickal9dibdielaj5a8df8beg79chcgdhehgfhjiklmn
	olhgdfnkhmfnegjgfnhmckomkininjljlkmokoiojnmlnmlooo
13:	0000000011111112222222333334444555566677899accdfgh
	134578ae34568bc345679d467ad78be689c7898ababbdeeijk
	2f6b9cjg9g7dakh8ahbeif5lbigl9ifamjffhcgdegfhikjlmn
	omkdhlnienimfljjclgnmkonckhmdjhenkgjiokoolnmnmlooo
14:	0000000011111112222222333334444555566677899accdfgh
	1345789c34568ad34567be467ad78be689c7898ababbdeeijk
	2h6beajg9f7bckh8agd9if5lbifl9jgamiffhcgdegfhjiklmn
	omidnflkenjglmikclmhnjoncjgmdkhenkhkjoioonmlnmlooo
15:	0000000011111112222222333334444555566677889abcdeil
	1345689b345679a34578ab467ad789e689c7bc9daeghffghjm
	2cg7afdi8dhfbjcf6eg9kc59ebhacbfdbag8ehcfdgijkijkkn
	lnkjohemknigomejinhomdmligoljhoklfonjmkmimnnnllloo
16:	0000000011111112222222333334444555566677889abcdeil
	134568ab345679b345789a467ae789c689d7be9cadfghfghjm
	2cf7g9dk8dgafich6ebfcj59dbgaebhcbaf8dfegchkijijkkn
	lnjihoemjnkohmeiknogdmmlkfoligojlhonimjmkmnnnllloo
17:	0000000011111112222222333333344445555666677778889a
	147adgjm4569chi4569bdf45689ae9bcf9ach89abbcdeabicd
	258behkn78ebfkl87caekgdb7cfglgkehefgjfildfighegjdj
	369cfiloadgjmnonlohimjimjhoknmojlkinoknmhnkomolmln
18:	0000000011111112222222333333344445555666677778889a
	147adgjm4569cef4569bcd45689ae9bdf9abe89acbcdiabcfd
	258behkn78bhkil87jafghhc7jbigcekggfihelhdheglgkfij
	369cfiloadgjmnoikoemnlmlfndokolnjmnjomnkinjomlohkm

1.3 Parameter Tables

1.35 Table Admissible parameter sets of nontrivial BIBDs with r ≤ 41 and k ≤ v/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 ≤ k ≤ v/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 $\hat{B}$ ; then $(V, \hat{B} \cup D)$ 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	Nr Comments, Ref	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

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§II.2	BIBDs with block size 3.
§II.3	Existence results on designs with “small” block size.
§II.7	Resolvable designs.
§II.6	Symmetric designs.
§II.5	Steiner systems.
§IV.2	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.

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2-( v , k , λ) Designs of Small Order

Handbook of Combinatorial Designs

Abstract

2-( v , k , λ) Designs of Small Order

1.1 Definition and Basics

1.2 Small Examples

1.3 Parameter Tables

Related chapters

Finite ...

Steiner ...

Difference ...

Divisible ...

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