1379:
1698:
4750:
1043:
1415:
1374:{\displaystyle {\begin{pmatrix}1&1&1&1&1&0&0&0&0&0\\1&1&0&0&0&1&1&1&0&0\\1&0&1&0&0&1&0&0&1&1\\0&1&0&1&0&0&1&0&1&1\\0&0&1&0&1&0&1&1&1&0\\0&0&0&1&1&1&0&1&0&1\\\end{pmatrix}}}
4975:
4474:
2070:= 2; that is, every set of two points is contained in two blocks ("lines"), while any two lines intersect in two points. They are similar to finite projective planes, except that rather than two points determining one line (and two lines determining one point), two points determine two lines (respectively, points). A biplane of order
1693:{\displaystyle \left({\begin{matrix}1&1&1&0&0&0&0\\1&0&0&1&1&0&0\\1&0&0&0&0&1&1\\0&1&0&1&0&1&0\\0&1&0&0&1&0&1\\0&0&1&1&0&0&1\\0&0&1&0&1&1&0\end{matrix}}\right)}
4745:{\displaystyle {\begin{pmatrix}1&1&1&1&0&0&0&0\\1&1&0&0&1&1&0&0\\0&0&1&1&1&1&0&0\\1&0&1&0&0&0&1&1\\0&1&0&0&1&0&1&1\\0&0&0&1&0&1&1&1\\\end{pmatrix}}}
4759:
978:
given block) constant. For other designs such as partially balanced incomplete block designs this may however be possible. Many such cases are discussed in. However, it can also be observed trivially for the magic squares or magic rectangles which can be viewed as the partially balanced incomplete block designs.
5420:
Suppose that skin cancer researchers want to test three different sunscreens. They coat two different sunscreens on the upper sides of the hands of a test person. After a UV radiation they record the skin irritation in terms of sunburn. The number of treatments is 3 (sunscreens) and the block size is
3275:
Note that the projective plane of order two is an
Hadamard 2-design; the projective plane of order four has parameters which fall in case 2; the only other known symmetric 2-designs with parameters in case 2 are the order 9 biplanes, but none of them are extendable; and there is no known symmetric
977:
A rather surprising and not very obvious (but very general) combinatorial result for these designs is that if points are denoted by any arbitrarily chosen set of equally or unequally spaced numerics, there is no choice of such a set which can make all block-sums (that is, sum of all points in a
2731:
4970:{\displaystyle {\begin{pmatrix}4&2&2&2&1&1\\2&4&2&1&2&1\\2&2&4&1&1&2\\2&1&1&4&2&2\\1&2&1&2&4&2\\1&1&2&2&2&4\\\end{pmatrix}}}
1388:
0123 0124 0156 0257 0345 0367 0467 1267 1346 1357 1457 2347 2356
2006:
of the projective plane. There can be no repeated lines since λ = 1, so a projective plane is a simple 2-design in which the number of lines and the number of points are always the same. For a projective plane,
2963:
5388:
While the origins of the subject are grounded in biological applications (as is some of the existing terminology), the designs are used in many applications where systematic comparisons are being made, such as in
2049:
Projective planes are known to exist for all orders which are prime numbers or powers of primes. They form the only known infinite family (with respect to having a constant λ value) of symmetric block designs.
2843:
5173:
6849:: Databases of combinatorial, statistical, and experimental block designs. Software and other resources hosted by the School of Mathematical Sciences at Queen Mary College, University of London.
5320:
5243:
2585:
3700:
5856:. These alternatives have been used in an attempt to replace the term "symmetric", since there is nothing symmetric (in the usual meaning of the term) about these designs. The use of
5093:
5872:, Cambridge, 1991) and captures the implication of v = b on the incidence matrix. Neither term has caught on as a replacement and these designs are still universally referred to as
3616:
3998:
1828:
788:
2109:(Trivial) The order 0 biplane has 2 points (and lines of size 2; a 2-(2,2,2) design); it is two points, with two blocks, each consisting of both points. Geometrically, it is the
3879:
3834:
3763:
1924:
3914:
1010:
012 013 024 035 045 125 134 145 234 235.
3789:
707:
5032:
389:
2379:
2350:
5686:
It is impossible to use a complete design (all treatments in each block) in this example because there are 3 sunscreens to test, but only 2 hands on each person.
2405:
1393:
The unique (7,3,1)-design is symmetric and has 7 blocks with each element repeated 3 times. Using the symbols 0 − 6, the blocks are the following triples:
1384:
One of four nonisomorphic (8,4,3)-designs has 14 blocks with each element repeated 7 times. Using the symbols 0 − 7 the blocks are the following 4-tuples:
5341:
PBIBD(2)s have been studied the most since they are the simplest and most useful of the PBIBDs. They fall into six types based on a classification of the
307:
design only when the design is also binary. The incidence matrix of a non-binary design lists the number of times each element is repeated in each block.
7095:
2410:
This construction is reversible, and the incidence matrix of a symmetric 2-design with these parameters can be used to form an
Hadamard matrix of size 4
6715:
6684:
17:
1409:
to the points and lines of the plane. Its corresponding incidence matrix can also be symmetric, if the labels or blocks are sorted the right way:
2848:
6903:
5789:
of order six. The 2-design with the indicated parameters is equivalent to the existence of five mutually orthogonal Latin squares of order six.
3325: + 1 points, no three collinear. It can be shown that every plane (which is a hyperplane since the geometric dimension is 3) of PG(3,
3285:
2760:
543:, so that no block contains all the elements of the set. This is the meaning of "incomplete" in the name of these designs.) In a table:
261:. The designs discussed in this article are all uniform. Block designs that are not necessarily uniform have also been studied; for
6370:; Shimamoto, T. (1952), "Classification and analysis of partially balanced incomplete block designs with two associate classes",
6830:
6640:
6617:
6595:
6556:
6425:
6333:
5679:) contains the treatments 1 and 2 simultaneously and the same applies to the pairs of treatments (1,3) and (2,3). Therefore,
5099:
6801:
2284:(that is, converted to an equivalent Hadamard matrix) where the first row and first column entries are all +1. If the size
7064:
1708:
The case of equality in Fisher's inequality, that is, a 2-design with an equal number of points and blocks, is called a
7110:
7054:
6896:
1840:
7024:
2726:{\displaystyle \lambda _{i}=\lambda \left.{\binom {v-i}{t-i}}\right/{\binom {k-i}{t-i}}{\text{ for }}i=0,1,\ldots ,t,}
1843:
gives necessary, but not sufficient, conditions for the existence of a symmetric design in terms of these parameters.
1731:, and, while it is generally not true in arbitrary 2-designs, in a symmetric design every two distinct blocks meet in
27:
This article is about block designs with fixed block size (uniform). For block designs with variable block sizes, see
6811:
6784:
6682:
Khattree, Ravindra (2019). "A note on the nonexistence of the constant block-sum balanced incomplete block designs".
6574:
6497:
6475:
6403:
6345:
6302:
5749:
5249:
5179:
2179:
266:
3311:
It is possible to give a geometric description of some inversive planes, indeed, of all known inversive planes. An
7100:
2299:
in standardized form, remove the first row and first column and convert every −1 to a 0. The resulting 0–1 matrix
76:, chosen such that frequency of the elements satisfies certain conditions making the collection of blocks exhibit
7131:
6753:
6659:
6270:
3628:
2168:
1397:
013 026 045 124 156 235 346.
7167:
1712:. Symmetric designs have the smallest number of blocks among all the 2-designs with the same number of points.
5038:
7157:
7074:
6889:
6856:
6434:
Fisher, R.A. (1940), "An examination of the different possible solutions of a problem in incomplete blocks",
3357:
3558:
2159:
1860:
7105:
7090:
3947:
2116:
The order 1 biplane has 4 points (and lines of size 3; a 2-(4,3,2) design); it is the complete design with
1774:
734:
409:
5556:
blocks, that is, 3 test people in order to obtain a balanced incomplete block design. Labeling the blocks
1759:-element subsets (the "blocks"), such that any two distinct blocks have exactly λ points in common, then (
5538:
for the block design which are then inserted into the R-function. Subsequently, the remaining parameters
3422:
3313:
2430:), each of which forms a partition of the point set of the BIBD. The set of parallel classes is called a
3439:
is odd, then any ovoid is projectively equivalent to the elliptic quadric in a projective geometry PG(3,
6324:
2464:
Consequently, a symmetric design can not have a non-trivial (more than one parallel class) resolution.
2213:
7162:
6635:, Carus Mathematical Monographs, vol. 14, Mathematical Association of America, pp. 96–130,
5409:
3132:) design. Note that derived designs with respect to different points may not be isomorphic. A design
3839:
3794:
3723:
7006:
6205:
5425:
340:
7039:
2468:
2196:
1879:
40:
3884:
2138:
The order 3 biplane has 11 points (and lines of size 5; a 2-(11,5,2)), and is also known as the
7152:
2191:
There are three biplanes of order 4 (and 16 points, lines of size 6; a 2-(16,6,2)). One is the
959:
7049:
6182:
Not a mathematical classification since one of the types is a catch-all "and everything else".
2124:= 3. Geometrically, the points are the vertices of a tetrahedron and the blocks are its faces.
272:
Block designs may or may not have repeated blocks. Designs without repeated blocks are called
7126:
6994:
6748:
6628:
5405:
5378:
2472:
296:
124:
32:
5950:
3768:
677:
47:
7059:
7029:
6969:
6947:
6713:
Khattree, Ravindra (2022). "On construction of equireplicated constant block-sum designs".
6531:
6227:
5382:
5005:
2192:
1406:
362:
28:
2355:
2326:
872:. These conditions are not sufficient as, for example, a (43,7,1)-design does not exist.
123:
which has been the most intensely studied type historically due to its application in the
8:
7069:
7034:
7014:
6912:
6852:
6793:
6605:
6583:
2384:
328:
299:). There, a design in which each element occurs the same total number of times is called
81:
65:
6231:
7044:
6964:
6732:
6701:
6535:
6449:
6436:
6414:
6392:
6243:
6217:
5695:
3491:
3476:
2131:: it has 7 points (and lines of size 4; a 2-(7,4,2)), where the lines are given as the
895:
of a 2-design is obtained by replacing each block with its complement in the point set
232:
101:
89:
6767:
6864:
6826:
6807:
6780:
6736:
6705:
6673:
6650:
6636:
6613:
6591:
6570:
6552:
6539:
6493:
6471:
6421:
6399:
6329:
6312:
6298:
6284:
6265:
6239:
5864:, Springer, 1968), in analogy with the most common example, projective planes, while
5768:
5377:
The mathematical subject of block designs originated in the statistical framework of
3619:
3480:
277:
2208:
Two biplanes are known of order 11 (and 79 points, lines of size 13; a 2-(79,13,2)).
2205:
There are five biplanes of order 9 (and 56 points, lines of size 11; a 2-(56,11,2)).
6974:
6926:
6762:
6744:
6724:
6693:
6668:
6519:
6486:
6453:
6445:
6383:
6379:
6354:
6279:
6247:
6235:
5758:
5599:
5397:
5390:
3204:
2304:
1855:
1015:
399:
93:
6728:
6697:
2202:
There are four biplanes of order 7 (and 37 points, lines of size 9; a 2-(37,9,2)).
6867:
6527:
5408:. The rows of their incidence matrices are also used as the symbols in a form of
3510:
2579:
and the four numbers themselves cannot be chosen arbitrarily. The equations are
2225:
406:
403:
85:
6751:(1970), "Non-isomorphic solutions of some balanced incomplete block designs I",
6506:
Kaski, Petteri; Östergård, Patric (2008). "There Are
Exactly Five Biplanes with
6208:(Jul 2012). "Expurgated PPM Using Symmetric Balanced Incomplete Block Designs".
4094:(3) be the following association scheme with three associate classes on the set
3301:
833:) even without assuming it explicitly, thus proving that the condition that any
402:
of a regular uniform block design. Also, each configuration has a corresponding
6957:
6797:
6343:(1949), "A Note on Fisher's Inequality for Balanced Incomplete Block Designs",
5700:
5433:
3407:
3042:
2154:
of order 11, which is constructed using the field with 11 elements, and is the
69:
6359:
5763:
5744:
295:
block designs, in which blocks may contain multiple copies of an element (see
7146:
6316:
5772:
2151:
2147:
1026:
963:
395:
54:
2023: + 1 is the number of lines with which a given point is incident.
5786:
5381:. These designs were especially useful in applications of the technique of
249:
A block design in which all the blocks have the same size (usually denoted
97:
2958:{\displaystyle r=\lambda _{1}=\lambda {v-1 \choose t-1}/{k-1 \choose t-1}}
1736:
419:
159:
is unspecified, it can usually be assumed to be 2, which means that each
57:
5785:
Proved by Tarry in 1900 who showed that there was no pair of orthogonal
3447:
is a prime power and there is a unique egglike inversive plane of order
7019:
6984:
6931:
6881:
6367:
6340:
5742:
5401:
2128:
2031:
1951: − 1 and, from the displayed equation above, we obtain
1402:
671:
are possible. The two basic equations connecting these parameters are
535:
of the design. (To avoid degenerate examples, it is also assumed that
413:
288:
36:
6523:
6458:
6872:
2038:= 4 + 2 + 1 = 7 points and 7 lines. In the Fano plane, each line has
1006:= 5). Using the symbols 0 − 5, the blocks are the following triples:
163:
of elements is found in the same number of blocks and the design is
80:(balance). Block designs have applications in many areas, including
2167:
Algebraically this corresponds to the exceptional embedding of the
332:
281:
77:
6222:
2838:{\displaystyle b=\lambda _{0}=\lambda {v \choose t}/{k \choose t}}
6588:
Constructions and
Combinatorial Problems in Design of Experiments
3257: = (λ + 2)(λ + 4λ + 2),
1871:> 1. For these designs the symmetric design equation becomes:
203:=1). When the balancing requirement fails, a design may still be
5385:. This remains a significant area for the use of block designs.
3011:. (Note that the "lambda value" changes as above and depends on
2212:
Biplanes of orders 5, 6, 8 and 10 do not exist, as shown by the
4082:) determines an association scheme but the converse is false.
3219: + 1, 1) designs) are those of orders 2 and 4.
2110:
1846:
The following are important examples of symmetric 2-designs:
3244:,λ) design, is extendable, then one of the following holds:
2426:
is a BIBD whose blocks can be partitioned into sets (called
821:
is a block that contains them both. This equation for every
343:. Such a design is uniform and regular: each block contains
242:, meaning that the collection of blocks is not all possible
5745:"On balanced incomplete-block designs with repeated blocks"
3421:
is an odd power of 2, another type of ovoid is known – the
2646:
2606:
2381:
points/blocks. Each pair of points is contained in exactly
310:
265:=2 they are known in the literature under the general name
6846:
2030:= 2 we get a projective plane of order 2, also called the
171:=1, each element occurs in the same number of blocks (the
151:-subsets of the original set occur in equally many (i.e.,
5908:
5906:
5400:
of block designs provide a natural source of interesting
4441: = 1. Also, for the association scheme we have
3451:. (But it is unknown if non-egglike ones exist.) (b) if
2011:
is the number of points on each line and it is equal to
6610:
Block
Designs: Analysis, Combinatorics and Applications
5168:{\displaystyle \sum _{i=1}^{m}n_{i}\lambda _{i}=r(k-1)}
2478:
46:"BIBD" redirects here. For the airport in Iceland, see
6862:
5903:
5743:
P. Dobcsányi, D.A. Preece. L.H. Soicher (2007-10-01).
4768:
4483:
4043:
blocks, such that there is an association scheme with
3466:
1839:, so the number of points is far from arbitrary. The
1424:
1052:
955:. A 2-design and its complement have the same order.
420:
Pairwise balanced uniform designs (2-designs or BIBDs)
6420:(2nd ed.), Boca Raton: Chapman & Hall/ CRC,
5252:
5182:
5102:
5041:
5008:
4762:
4477:
3950:
3887:
3842:
3797:
3771:
3726:
3631:
3561:
2851:
2763:
2588:
2387:
2358:
2329:
1978:
As a projective plane is a symmetric design, we have
1971: + 1 points in a projective plane of order
1882:
1777:
1418:
1046:
737:
680:
599:
number of blocks containing any 2 (or more generally
365:
6743:
5952:
From
Biplanes to the Klein quartic and the Buckyball
5549:
Using the basic relations we calculate that we need
3284:
A design with the parameters of the extension of an
2280:
identity matrix. An
Hadamard matrix can be put into
2042: + 1 = 3 points and each point belongs to
391:, which is the total number of element occurrences.
6266:"On collineation groups of symmetric block designs"
6203:
3463:is egglike (but there may be some unknown ovoids).
3345:
are the blocks of an inversive plane of order
3055:
643:)-design. The parameters are not all independent;
291:, the concept of a block design may be extended to
6649:Salwach, Chester J.; Mezzaroba, Joseph A. (1978).
6485:
6413:
6391:
6310:
6038:
5949:Martin, Pablo; Singerman, David (April 17, 2008),
5314:
5237:
5167:
5087:
5026:
4969:
4744:
3992:
3908:
3873:
3828:
3783:
3757:
3694:
3610:
3337: + 1 points. The plane sections of size
2957:
2837:
2725:
2399:
2373:
2344:
1918:
1822:
1692:
1373:
782:
701:
383:
6777:Combinatorial Designs: Constructions and Analysis
6716:Communications in Statistics - Theory and Methods
6685:Communications in Statistics - Theory and Methods
6604:
6412:Colbourn, Charles J.; Dinitz, Jeffrey H. (2007),
3459:is a power of 2 and any inversive plane of order
3349:. Any inversive plane arising this way is called
2949:
2920:
2906:
2877:
2829:
2816:
2802:
2789:
2682:
2653:
2640:
2611:
2571:,λ)-design. Again, these four numbers determine
2467:Archetypical resolvable 2-designs are the finite
452:, standing for balanced incomplete block design)
7144:
6648:
5968:
5570:, to avoid confusion, we have the block design,
6389:
6372:Journal of the American Statistical Association
6098:
6074:
5948:
5942:
5315:{\displaystyle n_{i}p_{jh}^{i}=n_{j}p_{ih}^{j}}
5238:{\displaystyle \sum _{u=0}^{m}p_{ju}^{h}=n_{j}}
2158:associated to the size 12 Hadamard matrix; see
1703:
6820:
6505:
6470:(2nd ed.), New York: Wiley-Interscience,
6411:
6158:
6110:
5979:
5836:
5718:
5325:A PBIBD(1) is a BIBD and a PBIBD(2) in which λ
3296: + 1, 1) design, is called a finite
860:must be integers, which imposes conditions on
195:-values), so for example a pairwise balanced (
6897:
6859:'s page of web based design theory resources.
6564:
6366:
5424:A corresponding BIBD can be generated by the
5346:
5336:
3222:Every Hadamard 2-design is extendable (to an
2352:blocks/points; each contains/is contained in
2127:The order 2 biplane is the complement of the
1766:The parameters of a symmetric design satisfy
1405:, with the elements and blocks of the design
6792:
6170:
6146:
6134:
6122:
3689:
3645:
3605:
3575:
3261: = λ + 3λ + 1,
3018:A consequence of this theorem is that every
1002:= 10) and each element is repeated 5 times (
899:. It is also a 2-design and has parameters
276:, in which case the "family" of blocks is a
246:-subsets, thus ruling out a trivial design.
238:Designs are usually said (or assumed) to be
6483:
6086:
6062:
6026:
6014:
5912:
4469: = 2. The incidence matrix M is
3695:{\displaystyle R^{*}:=\{(x,y)|(y,x)\in R\}}
3435:be a positive integer, at least 2. (a) If
2559:of the design. The design may be called a
849:can be computed from the other parameters.
6904:
6890:
6582:
6263:
6191:
5990:
5415:
4017:partially balanced incomplete block design
3353:. All known inversive planes are egglike.
2531:appears in exactly λ blocks. The numbers
712:obtained by counting the number of pairs (
579:number of blocks containing a given point
347:elements and each element is contained in
119:), specifically (and also synonymously) a
6766:
6672:
6492:, Cambridge: Cambridge University Press,
6457:
6358:
6297:, Cambridge: Cambridge University Press,
6292:
6283:
6221:
5936:
5762:
5436:and is specified in the following table:
3360:, the set of zeros of the quadratic form
3116:which contain p with p removed. It is a (
2743:is the number of blocks that contain any
2135:of the (3-point) lines in the Fano plane.
398:with constant row and column sums is the
6911:
6712:
6681:
6549:Symmetric Designs: An Algebraic Approach
6390:Cameron, P. J.; van Lint, J. H. (1991),
5820:
5809:
5798:
5520:The investigator chooses the parameters
5088:{\displaystyle \sum _{i=1}^{m}n_{i}=v-1}
3276:2-design with the parameters of case 3.
2475:is a resolution of a 2-(15,3,1) design.
2105:The 18 known examples are listed below.
480:blocks, and any pair of distinct points
315:The simplest type of "balanced" design (
311:Regular uniform designs (configurations)
187:is also balanced in all lower values of
107:Without further specifications the term
6774:
6050:
6002:
5885:
5730:
4071:, then they are together in precisely λ
2417:
2319: − 1) design called an
2155:
215:classes, each with its own (different)
14:
7145:
6546:
6433:
6394:Designs, Graphs, Codes and their Links
5870:Designs, Graphs, Codes and their Links
5665:Each treatment occurs in 2 blocks, so
3611:{\displaystyle R_{0}=\{(x,x):x\in X\}}
6885:
6863:
6821:van Lint, J.H.; Wilson, R.M. (1992).
6626:
5897:
5602:is specified in the following table:
4413:The parameters of this PBIBD(3) are:
4012:. Most authors assume this property.
3993:{\displaystyle p_{ij}^{k}=p_{ji}^{k}}
3112:} and as block set all the blocks of
2219:
1823:{\displaystyle \lambda (v-1)=k(k-1).}
783:{\displaystyle \lambda (v-1)=r(k-1),}
508:blocks is redundant, as shown below.
496:blocks. Here, the condition that any
6803:Combinatorics of Experimental Design
6565:Lindner, C.C.; Rodger, C.A. (1997),
6465:
6339:
5924:
5832:
5830:
5828:
3928:but not on the particular choice of
3052:by itself usually means a 2-design.
2046: + 1 = 3 lines.
1849:
1835:This imposes strong restrictions on
68:consisting of a set together with a
6608:; Padgett, L.V. (11 October 2005).
5848:They have also been referred to as
4039:and with each element appearing in
3467:Partially balanced designs (PBIBDs)
3279:
2180:projective linear group: action on
1401:This design is associated with the
793:obtained from counting for a fixed
351:blocks. The number of set elements
127:. Its generalization is known as a
24:
6484:Hughes, D.R.; Piper, E.C. (1985),
6450:10.1111/j.1469-1809.1940.tb02237.x
4374:The blocks of a PBIBD(3) based on
3525:. A pair of elements in relation R
2924:
2881:
2820:
2793:
2657:
2615:
2295:Given an Hadamard matrix of size 4
25:
7179:
6840:
6416:Handbook of Combinatorial Designs
6346:Annals of Mathematical Statistics
5825:
5750:European Journal of Combinatorics
4754:and the concurrence matrix MM is
3271: = 39, λ = 3.
6512:Journal of Combinatorial Designs
6293:Assmus, E.F.; Key, J.D. (1992),
6240:10.1109/LCOMM.2012.042512.120457
6039:Beth, Jungnickel & Lenz 1986
4027:)) is a block design based on a
3188:) design has an extension, then
3056:Derived and extendable t-designs
2094: + 1)/2 points (since
1998: + 1 also. The number
113:balanced incomplete block design
18:Balanced incomplete block design
6754:Journal of Combinatorial Theory
6660:Journal of Combinatorial Theory
6271:Journal of Combinatorial Theory
6197:
6185:
6176:
6164:
6152:
6140:
6128:
6116:
6104:
6092:
6080:
6068:
6056:
6044:
6032:
6020:
6008:
5996:
5984:
5973:
5962:
5930:
5918:
5891:
5879:
5372:
2244:whose entries are ±1 such that
2195:. These three designs are also
2169:projective special linear group
1763:) is a symmetric block design.
962:, named after the statistician
879:of a 2-design is defined to be
179:) and the design is said to be
7055:Cremona–Richmond configuration
6825:. Cambridge University Press.
6551:, Cambridge University Press,
6398:, Cambridge University Press,
6384:10.1080/01621459.1952.10501161
5842:
5814:
5803:
5792:
5779:
5736:
5724:
5712:
5546:are determined automatically.
5162:
5150:
4979:from which we can recover the
3874:{\displaystyle (z,y)\in R_{j}}
3855:
3843:
3829:{\displaystyle (x,z)\in R_{i}}
3810:
3798:
3758:{\displaystyle (x,y)\in R_{k}}
3739:
3727:
3680:
3668:
3664:
3660:
3648:
3590:
3578:
3356:An example of an ovoid is the
2082: + 2 points; it has
1910:
1898:
1814:
1802:
1793:
1781:
774:
762:
753:
741:
728:is a point in that block, and
659:, and not all combinations of
556:points, number of elements of
13:
1:
6768:10.1016/S0021-9800(70)80024-2
6729:10.1080/03610926.2020.1814816
6698:10.1080/03610926.2018.1508715
6257:
4990:
2066:is a symmetric 2-design with
1863:are symmetric 2-designs with
211:-subsets can be divided into
7132:Kirkman's schoolgirl problem
7065:Grünbaum–Rigby configuration
6775:Stinson, Douglas R. (2003),
6674:10.1016/0097-3165(78)90002-X
6627:Ryser, Herbert John (1963),
6285:10.1016/0097-3165(71)90054-9
6264:Aschbacher, Michael (1971).
5969:Salwach & Mezzaroba 1978
5383:analysis of variance (ANOVA)
3552:th associates. Furthermore:
2471:. A solution of the famous
2086: = 1 + (
2015: + 1. Similarly,
1704:Symmetric 2-designs (SBIBDs)
829:is constant (independent of
589:number of points in a block
199:=2) design is also regular (
183:. Any design balanced up to
7:
7025:Möbius–Kantor configuration
6466:Hall, Marshall Jr. (1986),
6210:IEEE Communications Letters
6099:Cameron & van Lint 1991
6075:Cameron & van Lint 1991
5689:
5347:Bose & Shimamoto (1952)
2747:-element set of points and
2535:(the number of elements of
2487:Given any positive integer
2053:
1941:order of a projective plane
1919:{\displaystyle v-1=k(k-1).}
1739:provides the converse. If
986:The unique (6,3,2)-design (
981:
515:(the number of elements of
134:
10:
7184:
7111:Bruck–Ryser–Chowla theorem
6629:"8. Combinatorial Designs"
6325:Cambridge University Press
6161:, pg. 562, Remark 42.3 (4)
6159:Colbourn & Dinitz 2007
6111:Colbourn & Dinitz 2007
6077:, pg. 11, Proposition 1.34
5980:Kaski & Östergård 2008
5837:Colbourn & Dinitz 2007
5719:Colbourn & Dinitz 2007
5337:Two associate class PBIBDs
4995:The parameters of a PBIBD(
4085:
3909:{\displaystyle p_{ij}^{k}}
3406:where f is an irreducible
2479:General balanced designs (
2445:,λ) resolvable design has
2214:Bruck-Ryser-Chowla theorem
2150:; it is associated to the
1853:
1841:Bruck–Ryser–Chowla theorem
412:known as its incidence or
45:
26:
7119:
7101:Szemerédi–Trotter theorem
7083:
7005:
6940:
6919:
6823:A Course in Combinatorics
6633:Combinatorial Mathematics
6569:, Boca Raton: CRC Press,
5764:10.1016/j.ejc.2006.08.007
5410:pulse-position modulation
4023:associate classes (PBIBD(
3940:An association scheme is
3410:in two variables over GF(
2292:must be a multiple of 4.
2074:is one whose blocks have
355:and the number of blocks
267:pairwise balanced designs
7091:Sylvester–Gallai theorem
6651:"The four biplanes with
6171:Street & Street 1987
6147:Street & Street 1987
6135:Street & Street 1987
6123:Street & Street 1987
6029:, pg. 158, Corollary 5.5
5706:
3251:is an Hadamard 2-design,
3166:if it has an extension.
3148:has a point p such that
3026:≥ 2 is also a 2-design.
2543:(the number of blocks),
2511:, such that every point
1861:Finite projective planes
1033:and constant column sum
845:blocks is redundant and
817:are distinct points and
611:The design is called a (
523:(the number of blocks),
341:Configuration (geometry)
231:, whose classes form an
7096:De Bruijn–Erdős theorem
7040:Desargues configuration
6853:Design Theory Resources
6360:10.1214/aoms/1177729958
6295:Designs and Their Codes
6113:, pg. 114, Remarks 6.35
6087:Hughes & Piper 1985
6063:Hughes & Piper 1985
6053:, pg.203, Corollary 9.6
6027:Hughes & Piper 1985
6015:Hughes & Piper 1985
5913:Hughes & Piper 1985
5860:is due to P.Dembowski (
5416:Statistical application
3192: + 1 divides
3120: − 1)-(
3041:,1)-design is called a
2449:parallel classes, then
2311: − 1, 2
2288: > 2 then
958:A fundamental theorem,
191:(though with different
139:A design is said to be
41:randomized block design
6749:Bhat-Nayak, Vasanti N.
6547:Lander, E. S. (1983),
6101:, pg. 11, Theorem 1.35
6089:, pg. 132, Theorem 4.5
6017:, pg. 156, Theorem 5.4
5868:is due to P. Cameron (
5421:2 (hands per person).
5406:error correcting codes
5365:partial geometry type;
5316:
5239:
5203:
5169:
5123:
5089:
5062:
5028:
4971:
4746:
4098:= {1,2,3,4,5,6}. The (
3994:
3910:
3875:
3830:
3785:
3784:{\displaystyle z\in X}
3759:
3696:
3612:
3128: − 1,
3124: − 1,
2983:,λ)-design is also an
2959:
2839:
2727:
2461: − 1.
2401:
2375:
2346:
2315: − 1,
1920:
1824:
1715:In a symmetric design
1694:
1375:
1029:with constant row sum
1014:and the corresponding
784:
703:
702:{\displaystyle bk=vr,}
385:
321:tactical configuration
223:=2 these are known as
7168:Design of experiments
7127:Design of experiments
6005:, pg. 74, Theorem 4.5
5937:Assmus & Key 1992
5379:design of experiments
5317:
5240:
5183:
5170:
5103:
5090:
5042:
5029:
5027:{\displaystyle vr=bk}
4972:
4747:
4429: = 4 and λ
3995:
3911:
3876:
3831:
3786:
3760:
3697:
3613:
2960:
2840:
2728:
2473:15 schoolgirl problem
2402:
2376:
2347:
1921:
1825:
1735:points. A theorem of
1695:
1376:
785:
704:
386:
384:{\displaystyle bk=vr}
335:is known simply as a
297:blocking (statistics)
125:design of experiments
7158:Combinatorial design
7060:Kummer configuration
7030:Pappus configuration
6913:Incidence structures
6794:Street, Anne Penfold
6612:. World Scientific.
6606:Raghavarao, Damaraju
6584:Raghavarao, Damaraju
6468:Combinatorial Theory
6206:Brandt-Pearce, Maïté
6041:, pg. 40 Example 5.8
5888:, pg.23, Theorem 2.2
5250:
5180:
5100:
5039:
5006:
4760:
4475:
4455: = 1 and
4437: = 2 and λ
4035:blocks each of size
3948:
3885:
3840:
3795:
3769:
3724:
3629:
3559:
3203:The only extendable
3108: − {
2849:
2761:
2586:
2503:-element subsets of
2418:Resolvable 2-designs
2385:
2374:{\displaystyle 2a-1}
2356:
2345:{\displaystyle 4a-1}
2327:
2259:is the transpose of
2193:Kummer configuration
2160:Paley construction I
1880:
1775:
1416:
1044:
998:= 2) has 10 blocks (
735:
678:
460:-element subsets of
428:(of elements called
363:
327:. The corresponding
111:usually refers to a
33:experimental designs
29:Combinatorial design
7070:Klein configuration
7050:Schläfli double six
7035:Hesse configuration
7015:Complete quadrangle
6232:2012arXiv1203.5378N
5434:R-package agricolae
5311:
5280:
5221:
4063:th associates, 1 ≤
4051:where, if elements
4047:classes defined on
3989:
3968:
3905:
3003:with 1 ≤
2519:appears in exactly
2424:resolvable 2-design
2400:{\displaystyle a-1}
2307:of a symmetric 2-(4
960:Fisher's inequality
887: −
424:Given a finite set
329:incidence structure
82:experimental design
66:incidence structure
7045:Reye configuration
6865:Weisstein, Eric W.
6798:Street, Deborah J.
6437:Annals of Eugenics
6313:Jungnickel, Dieter
6204:Noshad, Mohammad;
6149:, pg. 240, Lemma 4
5850:projective designs
5696:Incidence geometry
5359:Latin square type;
5312:
5294:
5263:
5235:
5204:
5165:
5085:
5024:
4967:
4961:
4742:
4736:
3990:
3972:
3951:
3906:
3888:
3871:
3826:
3781:
3755:
3692:
3618:and is called the
3608:
3537:. Each element of
3477:association scheme
3341: + 1 of
3267: = 495,
2955:
2835:
2723:
2523:blocks, and every
2397:
2371:
2342:
2220:Hadamard 2-designs
1916:
1820:
1747:-element set, and
1690:
1684:
1371:
1365:
780:
699:
651:, and λ determine
603:) distinct points
519:, called points),
456:to be a family of
381:
319:=1) is known as a
233:association scheme
205:partially balanced
173:replication number
102:algebraic geometry
90:physical chemistry
48:Bíldudalur Airport
7140:
7139:
6832:978-0-521-41057-1
6806:. Oxford U. P. .
6692:(20): 5165–5168.
6642:978-1-61444-014-7
6619:978-981-4480-23-9
6597:978-0-486-65685-4
6558:978-0-521-28693-0
6524:10.1002/jcd.20145
6427:978-1-58488-506-1
6334:978-0-521-44432-3
6328:. 2nd ed. (1999)
5862:Finite Geometries
5663:
5662:
5518:
5517:
5404:that are used as
4425: = 3,
4421: = 8,
4417: = 6,
4411:
4410:
4372:
4371:
4118:are in relation R
3620:Identity relation
3423:Suzuki–Tits ovoid
3329:) meets an ovoid
3236:, a symmetric 2-(
3224:Hadamard 3-design
3205:projective planes
3200: + 1).
3155:is isomorphic to
2999:)-design for any
2947:
2904:
2827:
2800:
2691:
2680:
2638:
2321:Hadamard 2-design
2282:standardized form
2156:Hadamard 2-design
2002:is the number of
1963: + 1 =
1939:we can write the
1850:Projective planes
1723:holds as well as
974:in any 2-design.
825:also proves that
607:
606:
569:number of blocks
444:≥ 1, we define a
165:pairwise balanced
70:family of subsets
16:(Redirected from
7175:
7163:Families of sets
6975:Projective plane
6927:Incidence matrix
6906:
6899:
6892:
6883:
6882:
6878:
6877:
6847:DesignTheory.Org
6836:
6817:
6789:
6771:
6770:
6745:Shrikhande, S.S.
6740:
6723:(2): 4434–4450.
6709:
6678:
6676:
6645:
6623:
6601:
6579:
6561:
6543:
6502:
6491:
6480:
6462:
6461:
6430:
6419:
6408:
6397:
6386:
6378:(258): 151–184,
6368:Bose, R. C.
6363:
6362:
6327:
6307:
6289:
6287:
6252:
6251:
6225:
6201:
6195:
6189:
6183:
6180:
6174:
6168:
6162:
6156:
6150:
6144:
6138:
6132:
6126:
6120:
6114:
6108:
6102:
6096:
6090:
6084:
6078:
6072:
6066:
6060:
6054:
6048:
6042:
6036:
6030:
6024:
6018:
6012:
6006:
6000:
5994:
5988:
5982:
5977:
5971:
5966:
5960:
5959:
5957:
5946:
5940:
5934:
5928:
5922:
5916:
5910:
5901:
5895:
5889:
5883:
5877:
5846:
5840:
5834:
5823:
5818:
5812:
5807:
5801:
5796:
5790:
5783:
5777:
5776:
5766:
5757:(7): 1955–1970.
5740:
5734:
5728:
5722:
5716:
5682:
5678:
5675:Just one block (
5671:
5605:
5604:
5600:incidence matrix
5598:A corresponding
5593:
5586:
5580:},
5579:
5569:
5565:
5555:
5545:
5541:
5537:
5533:
5526:
5439:
5438:
5398:incidence matrix
5391:software testing
5353:group divisible;
5321:
5319:
5318:
5313:
5310:
5305:
5293:
5292:
5279:
5274:
5262:
5261:
5244:
5242:
5241:
5236:
5234:
5233:
5220:
5215:
5202:
5197:
5174:
5172:
5171:
5166:
5143:
5142:
5133:
5132:
5122:
5117:
5094:
5092:
5091:
5086:
5072:
5071:
5061:
5056:
5033:
5031:
5030:
5025:
4976:
4974:
4973:
4968:
4966:
4965:
4751:
4749:
4748:
4743:
4741:
4740:
4407: 456
4393: 456
4381:
4380:
4368:
4363:
4358:
4353:
4348:
4343:
4331:
4326:
4321:
4316:
4311:
4306:
4294:
4289:
4284:
4279:
4274:
4269:
4257:
4252:
4247:
4242:
4237:
4232:
4220:
4215:
4210:
4205:
4200:
4195:
4183:
4178:
4173:
4168:
4163:
4158:
4125:
4124:
3999:
3997:
3996:
3991:
3988:
3983:
3967:
3962:
3915:
3913:
3912:
3907:
3904:
3899:
3880:
3878:
3877:
3872:
3870:
3869:
3835:
3833:
3832:
3827:
3825:
3824:
3790:
3788:
3787:
3782:
3765:, the number of
3764:
3762:
3761:
3756:
3754:
3753:
3701:
3699:
3698:
3693:
3667:
3641:
3640:
3617:
3615:
3614:
3609:
3571:
3570:
3511:binary relations
3490:together with a
3358:elliptic quadric
3304:, of order
3292: + 1,
3280:Inversive planes
3215: + 1,
2964:
2962:
2961:
2956:
2954:
2953:
2952:
2946:
2935:
2923:
2916:
2911:
2910:
2909:
2903:
2892:
2880:
2867:
2866:
2844:
2842:
2841:
2836:
2834:
2833:
2832:
2819:
2812:
2807:
2806:
2805:
2792:
2779:
2778:
2732:
2730:
2729:
2724:
2692:
2689:
2687:
2686:
2685:
2679:
2668:
2656:
2649:
2645:
2644:
2643:
2637:
2626:
2614:
2598:
2597:
2527:-element subset
2428:parallel classes
2406:
2404:
2403:
2398:
2380:
2378:
2377:
2372:
2351:
2349:
2348:
2343:
2305:incidence matrix
2144:
2143:
2090: + 2)(
2064:biplane geometry
1925:
1923:
1922:
1917:
1856:Projective plane
1829:
1827:
1826:
1821:
1755:-element set of
1710:symmetric design
1699:
1697:
1696:
1691:
1689:
1685:
1380:
1378:
1377:
1372:
1370:
1369:
1016:incidence matrix
841:is contained in
789:
787:
786:
781:
708:
706:
705:
700:
548:
547:
531:, and λ are the
504:is contained in
492:is contained in
476:is contained in
468:, such that any
400:incidence matrix
390:
388:
387:
382:
303:which implies a
94:software testing
21:
7183:
7182:
7178:
7177:
7176:
7174:
7173:
7172:
7143:
7142:
7141:
7136:
7115:
7079:
7001:
6936:
6932:Incidence graph
6915:
6910:
6868:"Block Designs"
6843:
6833:
6814:
6787:
6643:
6620:
6598:
6577:
6559:
6500:
6478:
6428:
6406:
6305:
6260:
6255:
6202:
6198:
6192:Raghavarao 1988
6190:
6186:
6181:
6177:
6169:
6165:
6157:
6153:
6145:
6141:
6133:
6129:
6121:
6117:
6109:
6105:
6097:
6093:
6085:
6081:
6073:
6069:
6061:
6057:
6049:
6045:
6037:
6033:
6025:
6021:
6013:
6009:
6001:
5997:
5991:Aschbacher 1971
5989:
5985:
5978:
5974:
5967:
5963:
5955:
5947:
5943:
5935:
5931:
5923:
5919:
5911:
5904:
5896:
5892:
5884:
5880:
5847:
5843:
5835:
5826:
5819:
5815:
5808:
5804:
5797:
5793:
5784:
5780:
5741:
5737:
5729:
5725:
5717:
5713:
5709:
5692:
5680:
5676:
5666:
5588:
5581:
5574:
5567:
5557:
5550:
5543:
5539:
5535:
5528:
5521:
5418:
5375:
5339:
5332:
5329: = λ
5328:
5306:
5298:
5288:
5284:
5275:
5267:
5257:
5253:
5251:
5248:
5247:
5229:
5225:
5216:
5208:
5198:
5187:
5181:
5178:
5177:
5138:
5134:
5128:
5124:
5118:
5107:
5101:
5098:
5097:
5067:
5063:
5057:
5046:
5040:
5037:
5036:
5007:
5004:
5003:
4993:
4977:
4960:
4959:
4954:
4949:
4944:
4939:
4934:
4928:
4927:
4922:
4917:
4912:
4907:
4902:
4896:
4895:
4890:
4885:
4880:
4875:
4870:
4864:
4863:
4858:
4853:
4848:
4843:
4838:
4832:
4831:
4826:
4821:
4816:
4811:
4806:
4800:
4799:
4794:
4789:
4784:
4779:
4774:
4764:
4763:
4761:
4758:
4757:
4752:
4735:
4734:
4729:
4724:
4719:
4714:
4709:
4704:
4699:
4693:
4692:
4687:
4682:
4677:
4672:
4667:
4662:
4657:
4651:
4650:
4645:
4640:
4635:
4630:
4625:
4620:
4615:
4609:
4608:
4603:
4598:
4593:
4588:
4583:
4578:
4573:
4567:
4566:
4561:
4556:
4551:
4546:
4541:
4536:
4531:
4525:
4524:
4519:
4514:
4509:
4504:
4499:
4494:
4489:
4479:
4478:
4476:
4473:
4472:
4468:
4461:
4454:
4447:
4440:
4436:
4432:
4404: 236
4401: 136
4398: 125
4390: 235
4387: 134
4384: 124
4366:
4361:
4356:
4351:
4346:
4341:
4329:
4324:
4319:
4314:
4309:
4304:
4292:
4287:
4282:
4277:
4272:
4267:
4255:
4250:
4245:
4240:
4235:
4230:
4218:
4213:
4208:
4203:
4198:
4193:
4181:
4176:
4171:
4166:
4161:
4156:
4121:
4088:
4074:
3984:
3976:
3963:
3955:
3949:
3946:
3945:
3900:
3892:
3886:
3883:
3882:
3865:
3861:
3841:
3838:
3837:
3820:
3816:
3796:
3793:
3792:
3770:
3767:
3766:
3749:
3745:
3725:
3722:
3721:
3663:
3636:
3632:
3630:
3627:
3626:
3566:
3562:
3560:
3557:
3556:
3547:
3529:are said to be
3528:
3524:
3520:
3516:
3469:
3397:
3390:
3379:
3373:
3333:in either 1 or
3298:inversive plane
3282:
3154:
3103:
3058:
2998:
2948:
2936:
2925:
2919:
2918:
2917:
2912:
2905:
2893:
2882:
2876:
2875:
2874:
2862:
2858:
2850:
2847:
2846:
2828:
2815:
2814:
2813:
2808:
2801:
2788:
2787:
2786:
2774:
2770:
2762:
2759:
2758:
2752:
2741:
2690: for
2688:
2681:
2669:
2658:
2652:
2651:
2650:
2639:
2627:
2616:
2610:
2609:
2608:
2605:
2593:
2589:
2587:
2584:
2583:
2485:
2434:of the design.
2420:
2386:
2383:
2382:
2357:
2354:
2353:
2328:
2325:
2324:
2323:. It contains
2271:
2254:
2226:Hadamard matrix
2222:
2141:
2140:
2056:
1986:, meaning that
1959: + 1)
1881:
1878:
1877:
1858:
1852:
1776:
1773:
1772:
1706:
1683:
1682:
1677:
1672:
1667:
1662:
1657:
1652:
1646:
1645:
1640:
1635:
1630:
1625:
1620:
1615:
1609:
1608:
1603:
1598:
1593:
1588:
1583:
1578:
1572:
1571:
1566:
1561:
1556:
1551:
1546:
1541:
1535:
1534:
1529:
1524:
1519:
1514:
1509:
1504:
1498:
1497:
1492:
1487:
1482:
1477:
1472:
1467:
1461:
1460:
1455:
1450:
1445:
1440:
1435:
1430:
1423:
1419:
1417:
1414:
1413:
1364:
1363:
1358:
1353:
1348:
1343:
1338:
1333:
1328:
1323:
1318:
1312:
1311:
1306:
1301:
1296:
1291:
1286:
1281:
1276:
1271:
1266:
1260:
1259:
1254:
1249:
1244:
1239:
1234:
1229:
1224:
1219:
1214:
1208:
1207:
1202:
1197:
1192:
1187:
1182:
1177:
1172:
1167:
1162:
1156:
1155:
1150:
1145:
1140:
1135:
1130:
1125:
1120:
1115:
1110:
1104:
1103:
1098:
1093:
1088:
1083:
1078:
1073:
1068:
1063:
1058:
1048:
1047:
1045:
1042:
1041:
984:
736:
733:
732:
724:is a block and
679:
676:
675:
623:)-design or a (
432:) and integers
422:
364:
361:
360:
359:are related by
313:
155:) blocks. When
137:
86:finite geometry
51:
44:
23:
22:
15:
12:
11:
5:
7181:
7171:
7170:
7165:
7160:
7155:
7138:
7137:
7135:
7134:
7129:
7123:
7121:
7117:
7116:
7114:
7113:
7108:
7106:Beck's theorem
7103:
7098:
7093:
7087:
7085:
7081:
7080:
7078:
7077:
7072:
7067:
7062:
7057:
7052:
7047:
7042:
7037:
7032:
7027:
7022:
7017:
7011:
7009:
7007:Configurations
7003:
7002:
7000:
6999:
6998:
6997:
6989:
6988:
6987:
6979:
6978:
6977:
6972:
6962:
6961:
6960:
6958:Steiner system
6955:
6944:
6942:
6938:
6937:
6935:
6934:
6929:
6923:
6921:
6920:Representation
6917:
6916:
6909:
6908:
6901:
6894:
6886:
6880:
6879:
6860:
6850:
6842:
6841:External links
6839:
6838:
6837:
6831:
6818:
6812:
6790:
6785:
6772:
6761:(2): 174–191,
6741:
6710:
6679:
6667:(2): 141–145.
6646:
6641:
6624:
6618:
6602:
6596:
6580:
6575:
6562:
6557:
6544:
6518:(2): 117–127.
6503:
6498:
6481:
6476:
6463:
6431:
6426:
6409:
6404:
6387:
6364:
6353:(4): 619–620,
6337:
6317:Lenz, Hanfried
6311:Beth, Thomas;
6308:
6303:
6290:
6278:(3): 272–281.
6259:
6256:
6254:
6253:
6216:(7): 968–971.
6196:
6184:
6175:
6163:
6151:
6139:
6127:
6115:
6103:
6091:
6079:
6067:
6055:
6043:
6031:
6019:
6007:
5995:
5983:
5972:
5961:
5941:
5929:
5917:
5902:
5890:
5878:
5854:square designs
5841:
5824:
5813:
5802:
5791:
5778:
5735:
5723:
5710:
5708:
5705:
5704:
5703:
5701:Steiner system
5698:
5691:
5688:
5661:
5660:
5657:
5654:
5651:
5647:
5646:
5643:
5640:
5637:
5633:
5632:
5629:
5626:
5623:
5619:
5618:
5615:
5612:
5609:
5596:
5595:
5516:
5515:
5512:
5509:
5505:
5504:
5501:
5498:
5494:
5493:
5490:
5487:
5483:
5482:
5479:
5476:
5472:
5471:
5468:
5465:
5461:
5460:
5457:
5454:
5450:
5449:
5446:
5443:
5417:
5414:
5374:
5371:
5370:
5369:
5368:miscellaneous.
5366:
5363:
5360:
5357:
5354:
5338:
5335:
5330:
5326:
5323:
5322:
5309:
5304:
5301:
5297:
5291:
5287:
5283:
5278:
5273:
5270:
5266:
5260:
5256:
5245:
5232:
5228:
5224:
5219:
5214:
5211:
5207:
5201:
5196:
5193:
5190:
5186:
5175:
5164:
5161:
5158:
5155:
5152:
5149:
5146:
5141:
5137:
5131:
5127:
5121:
5116:
5113:
5110:
5106:
5095:
5084:
5081:
5078:
5075:
5070:
5066:
5060:
5055:
5052:
5049:
5045:
5034:
5023:
5020:
5017:
5014:
5011:
4992:
4989:
4964:
4958:
4955:
4953:
4950:
4948:
4945:
4943:
4940:
4938:
4935:
4933:
4930:
4929:
4926:
4923:
4921:
4918:
4916:
4913:
4911:
4908:
4906:
4903:
4901:
4898:
4897:
4894:
4891:
4889:
4886:
4884:
4881:
4879:
4876:
4874:
4871:
4869:
4866:
4865:
4862:
4859:
4857:
4854:
4852:
4849:
4847:
4844:
4842:
4839:
4837:
4834:
4833:
4830:
4827:
4825:
4822:
4820:
4817:
4815:
4812:
4810:
4807:
4805:
4802:
4801:
4798:
4795:
4793:
4790:
4788:
4785:
4783:
4780:
4778:
4775:
4773:
4770:
4769:
4767:
4756:
4739:
4733:
4730:
4728:
4725:
4723:
4720:
4718:
4715:
4713:
4710:
4708:
4705:
4703:
4700:
4698:
4695:
4694:
4691:
4688:
4686:
4683:
4681:
4678:
4676:
4673:
4671:
4668:
4666:
4663:
4661:
4658:
4656:
4653:
4652:
4649:
4646:
4644:
4641:
4639:
4636:
4634:
4631:
4629:
4626:
4624:
4621:
4619:
4616:
4614:
4611:
4610:
4607:
4604:
4602:
4599:
4597:
4594:
4592:
4589:
4587:
4584:
4582:
4579:
4577:
4574:
4572:
4569:
4568:
4565:
4562:
4560:
4557:
4555:
4552:
4550:
4547:
4545:
4542:
4540:
4537:
4535:
4532:
4530:
4527:
4526:
4523:
4520:
4518:
4515:
4513:
4510:
4508:
4505:
4503:
4500:
4498:
4495:
4493:
4490:
4488:
4485:
4484:
4482:
4471:
4466:
4462: =
4459:
4452:
4448: =
4445:
4438:
4434:
4433: = λ
4430:
4409:
4408:
4405:
4402:
4399:
4395:
4394:
4391:
4388:
4385:
4370:
4369:
4364:
4362: 1
4359:
4357: 1
4354:
4349:
4344:
4342: 3
4339:
4333:
4332:
4330: 1
4327:
4322:
4320: 1
4317:
4312:
4307:
4302:
4296:
4295:
4293: 1
4290:
4288: 1
4285:
4280:
4275:
4270:
4265:
4259:
4258:
4253:
4248:
4243:
4238:
4236: 1
4233:
4231: 1
4228:
4222:
4221:
4216:
4211:
4206:
4204: 1
4201:
4199: 0
4196:
4194: 1
4191:
4185:
4184:
4179:
4174:
4172: 2
4169:
4167: 1
4164:
4162: 1
4159:
4154:
4148:
4147:
4144:
4141:
4138:
4135:
4132:
4129:
4119:
4087:
4084:
4072:
3987:
3982:
3979:
3975:
3971:
3966:
3961:
3958:
3954:
3938:
3937:
3903:
3898:
3895:
3891:
3881:is a constant
3868:
3864:
3860:
3857:
3854:
3851:
3848:
3845:
3823:
3819:
3815:
3812:
3809:
3806:
3803:
3800:
3780:
3777:
3774:
3752:
3748:
3744:
3741:
3738:
3735:
3732:
3729:
3718:
3691:
3688:
3685:
3682:
3679:
3676:
3673:
3670:
3666:
3662:
3659:
3656:
3653:
3650:
3647:
3644:
3639:
3635:
3623:
3607:
3604:
3601:
3598:
3595:
3592:
3589:
3586:
3583:
3580:
3577:
3574:
3569:
3565:
3545:
3526:
3522:
3518:
3514:
3479:consists of a
3468:
3465:
3455:is even, then
3408:quadratic form
3404:
3403:
3402:
3401:
3400:
3399:
3395:
3388:
3377:
3371:
3321:) is a set of
3281:
3278:
3273:
3272:
3262:
3252:
3207:(symmetric 2-(
3152:
3104:has point set
3099:
3094:derived design
3057:
3054:
3043:Steiner system
2996:
2951:
2945:
2942:
2939:
2934:
2931:
2928:
2922:
2915:
2908:
2902:
2899:
2896:
2891:
2888:
2885:
2879:
2873:
2870:
2865:
2861:
2857:
2854:
2831:
2826:
2823:
2818:
2811:
2804:
2799:
2796:
2791:
2785:
2782:
2777:
2773:
2769:
2766:
2750:
2739:
2734:
2733:
2722:
2719:
2716:
2713:
2710:
2707:
2704:
2701:
2698:
2695:
2684:
2678:
2675:
2672:
2667:
2664:
2661:
2655:
2648:
2642:
2636:
2633:
2630:
2625:
2622:
2619:
2613:
2607:
2604:
2601:
2596:
2592:
2499:is a class of
2484:
2477:
2419:
2416:
2396:
2393:
2390:
2370:
2367:
2364:
2361:
2341:
2338:
2335:
2332:
2267:
2252:
2248: = m
2221:
2218:
2210:
2209:
2206:
2203:
2200:
2188:
2187:
2164:
2163:
2136:
2125:
2114:
2055:
2052:
1929:
1928:
1927:
1926:
1915:
1912:
1909:
1906:
1903:
1900:
1897:
1894:
1891:
1888:
1885:
1867:= 1 and order
1854:Main article:
1851:
1848:
1833:
1832:
1831:
1830:
1819:
1816:
1813:
1810:
1807:
1804:
1801:
1798:
1795:
1792:
1789:
1786:
1783:
1780:
1705:
1702:
1701:
1700:
1688:
1681:
1678:
1676:
1673:
1671:
1668:
1666:
1663:
1661:
1658:
1656:
1653:
1651:
1648:
1647:
1644:
1641:
1639:
1636:
1634:
1631:
1629:
1626:
1624:
1621:
1619:
1616:
1614:
1611:
1610:
1607:
1604:
1602:
1599:
1597:
1594:
1592:
1589:
1587:
1584:
1582:
1579:
1577:
1574:
1573:
1570:
1567:
1565:
1562:
1560:
1557:
1555:
1552:
1550:
1547:
1545:
1542:
1540:
1537:
1536:
1533:
1530:
1528:
1525:
1523:
1520:
1518:
1515:
1513:
1510:
1508:
1505:
1503:
1500:
1499:
1496:
1493:
1491:
1488:
1486:
1483:
1481:
1478:
1476:
1473:
1471:
1468:
1466:
1463:
1462:
1459:
1456:
1454:
1451:
1449:
1446:
1444:
1441:
1439:
1436:
1434:
1431:
1429:
1426:
1425:
1422:
1399:
1398:
1391:
1390:
1382:
1381:
1368:
1362:
1359:
1357:
1354:
1352:
1349:
1347:
1344:
1342:
1339:
1337:
1334:
1332:
1329:
1327:
1324:
1322:
1319:
1317:
1314:
1313:
1310:
1307:
1305:
1302:
1300:
1297:
1295:
1292:
1290:
1287:
1285:
1282:
1280:
1277:
1275:
1272:
1270:
1267:
1265:
1262:
1261:
1258:
1255:
1253:
1250:
1248:
1245:
1243:
1240:
1238:
1235:
1233:
1230:
1228:
1225:
1223:
1220:
1218:
1215:
1213:
1210:
1209:
1206:
1203:
1201:
1198:
1196:
1193:
1191:
1188:
1186:
1183:
1181:
1178:
1176:
1173:
1171:
1168:
1166:
1163:
1161:
1158:
1157:
1154:
1151:
1149:
1146:
1144:
1141:
1139:
1136:
1134:
1131:
1129:
1126:
1124:
1121:
1119:
1116:
1114:
1111:
1109:
1106:
1105:
1102:
1099:
1097:
1094:
1092:
1089:
1087:
1084:
1082:
1079:
1077:
1074:
1072:
1069:
1067:
1064:
1062:
1059:
1057:
1054:
1053:
1051:
1012:
1011:
983:
980:
951: − 2
852:The resulting
791:
790:
779:
776:
773:
770:
767:
764:
761:
758:
755:
752:
749:
746:
743:
740:
710:
709:
698:
695:
692:
689:
686:
683:
609:
608:
605:
604:
597:
591:
590:
587:
581:
580:
577:
571:
570:
567:
561:
560:
554:
421:
418:
380:
377:
374:
371:
368:
312:
309:
301:equireplicate,
280:rather than a
136:
133:
9:
6:
4:
3:
2:
7180:
7169:
7166:
7164:
7161:
7159:
7156:
7154:
7153:Combinatorics
7151:
7150:
7148:
7133:
7130:
7128:
7125:
7124:
7122:
7118:
7112:
7109:
7107:
7104:
7102:
7099:
7097:
7094:
7092:
7089:
7088:
7086:
7082:
7076:
7073:
7071:
7068:
7066:
7063:
7061:
7058:
7056:
7053:
7051:
7048:
7046:
7043:
7041:
7038:
7036:
7033:
7031:
7028:
7026:
7023:
7021:
7018:
7016:
7013:
7012:
7010:
7008:
7004:
6996:
6993:
6992:
6990:
6986:
6983:
6982:
6981:Graph theory
6980:
6976:
6973:
6971:
6968:
6967:
6966:
6963:
6959:
6956:
6954:
6951:
6950:
6949:
6948:Combinatorics
6946:
6945:
6943:
6939:
6933:
6930:
6928:
6925:
6924:
6922:
6918:
6914:
6907:
6902:
6900:
6895:
6893:
6888:
6887:
6884:
6875:
6874:
6869:
6866:
6861:
6858:
6857:Peter Cameron
6854:
6851:
6848:
6845:
6844:
6834:
6828:
6824:
6819:
6815:
6813:0-19-853256-3
6809:
6805:
6804:
6799:
6795:
6791:
6788:
6786:0-387-95487-2
6782:
6778:
6773:
6769:
6764:
6760:
6756:
6755:
6750:
6746:
6742:
6738:
6734:
6730:
6726:
6722:
6718:
6717:
6711:
6707:
6703:
6699:
6695:
6691:
6687:
6686:
6680:
6675:
6670:
6666:
6662:
6661:
6656:
6654:
6647:
6644:
6638:
6634:
6630:
6625:
6621:
6615:
6611:
6607:
6603:
6599:
6593:
6589:
6585:
6581:
6578:
6576:0-8493-3986-3
6572:
6568:
6567:Design Theory
6563:
6560:
6554:
6550:
6545:
6541:
6537:
6533:
6529:
6525:
6521:
6517:
6513:
6509:
6504:
6501:
6499:0-521-25754-9
6495:
6490:
6489:
6488:Design theory
6482:
6479:
6477:0-471-09138-3
6473:
6469:
6464:
6460:
6455:
6451:
6447:
6443:
6439:
6438:
6432:
6429:
6423:
6418:
6417:
6410:
6407:
6405:0-521-42385-6
6401:
6396:
6395:
6388:
6385:
6381:
6377:
6373:
6369:
6365:
6361:
6356:
6352:
6348:
6347:
6342:
6338:
6335:
6331:
6326:
6322:
6321:Design Theory
6318:
6314:
6309:
6306:
6304:0-521-41361-3
6300:
6296:
6291:
6286:
6281:
6277:
6273:
6272:
6267:
6262:
6261:
6249:
6245:
6241:
6237:
6233:
6229:
6224:
6219:
6215:
6211:
6207:
6200:
6193:
6188:
6179:
6172:
6167:
6160:
6155:
6148:
6143:
6136:
6131:
6124:
6119:
6112:
6107:
6100:
6095:
6088:
6083:
6076:
6071:
6064:
6059:
6052:
6047:
6040:
6035:
6028:
6023:
6016:
6011:
6004:
5999:
5993:, pp. 279–281
5992:
5987:
5981:
5976:
5970:
5965:
5954:
5953:
5945:
5938:
5933:
5926:
5921:
5914:
5909:
5907:
5900:, pp. 102–104
5899:
5894:
5887:
5882:
5875:
5871:
5867:
5863:
5859:
5855:
5851:
5845:
5838:
5833:
5831:
5829:
5822:
5821:Khattree 2022
5817:
5811:
5810:Khattree 2022
5806:
5800:
5799:Khattree 2019
5795:
5788:
5787:Latin squares
5782:
5774:
5770:
5765:
5760:
5756:
5752:
5751:
5746:
5739:
5732:
5727:
5720:
5715:
5711:
5702:
5699:
5697:
5694:
5693:
5687:
5684:
5673:
5669:
5658:
5655:
5652:
5649:
5648:
5644:
5641:
5638:
5635:
5634:
5630:
5627:
5624:
5621:
5620:
5616:
5613:
5610:
5607:
5606:
5603:
5601:
5591:
5584:
5577:
5573:
5572:
5571:
5564:
5560:
5553:
5547:
5531:
5524:
5513:
5510:
5507:
5506:
5502:
5499:
5496:
5495:
5491:
5488:
5485:
5484:
5480:
5477:
5474:
5473:
5469:
5466:
5463:
5462:
5458:
5455:
5452:
5451:
5447:
5444:
5441:
5440:
5437:
5435:
5431:
5427:
5422:
5413:
5411:
5407:
5403:
5399:
5394:
5392:
5386:
5384:
5380:
5367:
5364:
5361:
5358:
5355:
5352:
5351:
5350:
5348:
5345:PBIBD(2)s by
5344:
5334:
5307:
5302:
5299:
5295:
5289:
5285:
5281:
5276:
5271:
5268:
5264:
5258:
5254:
5246:
5230:
5226:
5222:
5217:
5212:
5209:
5205:
5199:
5194:
5191:
5188:
5184:
5176:
5159:
5156:
5153:
5147:
5144:
5139:
5135:
5129:
5125:
5119:
5114:
5111:
5108:
5104:
5096:
5082:
5079:
5076:
5073:
5068:
5064:
5058:
5053:
5050:
5047:
5043:
5035:
5021:
5018:
5015:
5012:
5009:
5002:
5001:
5000:
4998:
4988:
4986:
4982:
4962:
4956:
4951:
4946:
4941:
4936:
4931:
4924:
4919:
4914:
4909:
4904:
4899:
4892:
4887:
4882:
4877:
4872:
4867:
4860:
4855:
4850:
4845:
4840:
4835:
4828:
4823:
4818:
4813:
4808:
4803:
4796:
4791:
4786:
4781:
4776:
4771:
4765:
4755:
4737:
4731:
4726:
4721:
4716:
4711:
4706:
4701:
4696:
4689:
4684:
4679:
4674:
4669:
4664:
4659:
4654:
4647:
4642:
4637:
4632:
4627:
4622:
4617:
4612:
4605:
4600:
4595:
4590:
4585:
4580:
4575:
4570:
4563:
4558:
4553:
4548:
4543:
4538:
4533:
4528:
4521:
4516:
4511:
4506:
4501:
4496:
4491:
4486:
4480:
4470:
4465:
4458:
4451:
4444:
4428:
4424:
4420:
4416:
4406:
4403:
4400:
4397:
4396:
4392:
4389:
4386:
4383:
4382:
4379:
4377:
4367: 0
4365:
4360:
4355:
4352: 2
4350:
4347: 3
4345:
4340:
4338:
4335:
4334:
4328:
4325: 0
4323:
4318:
4315: 3
4313:
4310: 2
4308:
4305: 3
4303:
4301:
4298:
4297:
4291:
4286:
4283: 0
4281:
4278: 3
4276:
4273: 3
4271:
4268: 2
4266:
4264:
4261:
4260:
4256: 2
4254:
4251: 3
4249:
4246: 3
4244:
4241: 0
4239:
4234:
4229:
4227:
4224:
4223:
4219: 3
4217:
4214: 2
4212:
4209: 3
4207:
4202:
4197:
4192:
4190:
4187:
4186:
4182: 3
4180:
4177: 3
4175:
4170:
4165:
4160:
4157: 0
4155:
4153:
4150:
4149:
4145:
4142:
4139:
4136:
4133:
4130:
4127:
4126:
4123:
4117:
4113:
4109:
4105:
4101:
4097:
4093:
4083:
4081:
4076:
4070:
4066:
4062:
4058:
4054:
4050:
4046:
4042:
4038:
4034:
4030:
4026:
4022:
4018:
4013:
4011:
4007:
4003:
3985:
3980:
3977:
3973:
3969:
3964:
3959:
3956:
3952:
3943:
3935:
3931:
3927:
3923:
3919:
3916:depending on
3901:
3896:
3893:
3889:
3866:
3862:
3858:
3852:
3849:
3846:
3821:
3817:
3813:
3807:
3804:
3801:
3778:
3775:
3772:
3750:
3746:
3742:
3736:
3733:
3730:
3719:
3717:
3713:
3709:
3705:
3686:
3683:
3677:
3674:
3671:
3657:
3654:
3651:
3642:
3637:
3633:
3624:
3621:
3602:
3599:
3596:
3593:
3587:
3584:
3581:
3572:
3567:
3563:
3555:
3554:
3553:
3551:
3544:
3540:
3536:
3532:
3512:
3508:
3504:
3500:
3496:
3493:
3489:
3485:
3482:
3478:
3474:
3464:
3462:
3458:
3454:
3450:
3446:
3442:
3438:
3434:
3430:
3426:
3424:
3420:
3415:
3413:
3409:
3394:
3387:
3383:
3376:
3370:
3367:
3366:
3365:
3364:
3363:
3362:
3361:
3359:
3354:
3352:
3348:
3344:
3340:
3336:
3332:
3328:
3324:
3320:
3316:
3315:
3309:
3307:
3303:
3299:
3295:
3291:
3288:, i.e., a 3-(
3287:
3277:
3270:
3266:
3263:
3260:
3256:
3253:
3250:
3247:
3246:
3245:
3243:
3239:
3235:
3231:
3227:
3225:
3220:
3218:
3214:
3211: +
3210:
3206:
3201:
3199:
3195:
3191:
3187:
3183:
3179:
3175:
3171:
3167:
3165:
3162:
3158:
3151:
3147:
3143:
3139:
3136:is called an
3135:
3131:
3127:
3123:
3119:
3115:
3111:
3107:
3102:
3098:
3095:
3091:
3087:
3084:) design and
3083:
3079:
3075:
3071:
3067:
3063:
3053:
3051:
3046:
3044:
3040:
3036:
3032:
3027:
3025:
3022:-design with
3021:
3016:
3014:
3010:
3007: ≤
3006:
3002:
2994:
2990:
2986:
2982:
2978:
2974:
2970:
2966:
2943:
2940:
2937:
2932:
2929:
2926:
2913:
2900:
2897:
2894:
2889:
2886:
2883:
2871:
2868:
2863:
2859:
2855:
2852:
2824:
2821:
2809:
2797:
2794:
2783:
2780:
2775:
2771:
2767:
2764:
2755:
2753:
2746:
2742:
2720:
2717:
2714:
2711:
2708:
2705:
2702:
2699:
2696:
2693:
2676:
2673:
2670:
2665:
2662:
2659:
2634:
2631:
2628:
2623:
2620:
2617:
2602:
2599:
2594:
2590:
2582:
2581:
2580:
2578:
2574:
2570:
2566:
2562:
2558:
2554:
2550:
2546:
2542:
2538:
2534:
2530:
2526:
2522:
2518:
2514:
2510:
2506:
2502:
2498:
2494:
2490:
2482:
2476:
2474:
2470:
2469:affine planes
2465:
2462:
2460:
2457: +
2456:
2453: ≥
2452:
2448:
2444:
2440:
2435:
2433:
2429:
2425:
2415:
2413:
2408:
2394:
2391:
2388:
2368:
2365:
2362:
2359:
2339:
2336:
2333:
2330:
2322:
2318:
2314:
2310:
2306:
2302:
2298:
2293:
2291:
2287:
2283:
2279:
2276: ×
2275:
2270:
2266:
2262:
2258:
2251:
2247:
2243:
2239:
2235:
2231:
2227:
2217:
2215:
2207:
2204:
2201:
2198:
2197:Menon designs
2194:
2190:
2189:
2185:
2183:
2178:(2,11) – see
2177:
2173:
2170:
2166:
2165:
2161:
2157:
2153:
2152:Paley digraph
2149:
2148:Raymond Paley
2145:
2142:Paley biplane
2137:
2134:
2130:
2126:
2123:
2119:
2115:
2112:
2108:
2107:
2106:
2103:
2101:
2098: =
2097:
2093:
2089:
2085:
2081:
2078: =
2077:
2073:
2069:
2065:
2061:
2051:
2047:
2045:
2041:
2037:
2033:
2029:
2024:
2022:
2018:
2014:
2010:
2005:
2001:
1997:
1994: +
1993:
1989:
1985:
1981:
1976:
1974:
1970:
1967: +
1966:
1962:
1958:
1954:
1950:
1946:
1942:
1938:
1934:
1913:
1907:
1904:
1901:
1895:
1892:
1889:
1886:
1883:
1876:
1875:
1874:
1873:
1872:
1870:
1866:
1862:
1857:
1847:
1844:
1842:
1838:
1817:
1811:
1808:
1805:
1799:
1796:
1790:
1787:
1784:
1778:
1771:
1770:
1769:
1768:
1767:
1764:
1762:
1758:
1754:
1750:
1746:
1742:
1738:
1734:
1730:
1726:
1722:
1718:
1713:
1711:
1686:
1679:
1674:
1669:
1664:
1659:
1654:
1649:
1642:
1637:
1632:
1627:
1622:
1617:
1612:
1605:
1600:
1595:
1590:
1585:
1580:
1575:
1568:
1563:
1558:
1553:
1548:
1543:
1538:
1531:
1526:
1521:
1516:
1511:
1506:
1501:
1494:
1489:
1484:
1479:
1474:
1469:
1464:
1457:
1452:
1447:
1442:
1437:
1432:
1427:
1420:
1412:
1411:
1410:
1408:
1407:corresponding
1404:
1396:
1395:
1394:
1387:
1386:
1385:
1366:
1360:
1355:
1350:
1345:
1340:
1335:
1330:
1325:
1320:
1315:
1308:
1303:
1298:
1293:
1288:
1283:
1278:
1273:
1268:
1263:
1256:
1251:
1246:
1241:
1236:
1231:
1226:
1221:
1216:
1211:
1204:
1199:
1194:
1189:
1184:
1179:
1174:
1169:
1164:
1159:
1152:
1147:
1142:
1137:
1132:
1127:
1122:
1117:
1112:
1107:
1100:
1095:
1090:
1085:
1080:
1075:
1070:
1065:
1060:
1055:
1049:
1040:
1039:
1038:
1036:
1032:
1028:
1027:binary matrix
1025:
1021:
1017:
1009:
1008:
1007:
1005:
1001:
997:
993:
989:
979:
975:
973:
970: ≥
969:
965:
964:Ronald Fisher
961:
956:
954:
950:
947: +
946:
942:
938:
935: −
934:
930:
926:
923: −
922:
918:
914:
910:
906:
902:
898:
894:
890:
886:
882:
878:
873:
871:
867:
863:
859:
855:
850:
848:
844:
840:
836:
832:
828:
824:
820:
816:
812:
808:
804:
800:
797:the triples (
796:
777:
771:
768:
765:
759:
756:
750:
747:
744:
738:
731:
730:
729:
727:
723:
719:
715:
696:
693:
690:
687:
684:
681:
674:
673:
672:
670:
666:
662:
658:
654:
650:
646:
642:
638:
634:
630:
626:
622:
618:
614:
602:
598:
596:
593:
592:
588:
586:
583:
582:
578:
576:
573:
572:
568:
566:
563:
562:
559:
555:
553:
550:
549:
546:
545:
544:
542:
538:
534:
530:
526:
522:
518:
514:
509:
507:
503:
499:
495:
491:
487:
483:
479:
475:
471:
467:
463:
459:
455:
451:
447:
443:
439:
435:
431:
427:
417:
415:
411:
408:
405:
401:
397:
396:binary matrix
392:
378:
375:
372:
369:
366:
358:
354:
350:
346:
342:
338:
337:configuration
334:
330:
326:
322:
318:
308:
306:
302:
298:
294:
290:
285:
283:
279:
275:
270:
268:
264:
260:
256:
252:
247:
245:
241:
236:
234:
230:
228:
222:
218:
214:
210:
206:
202:
198:
194:
190:
186:
182:
178:
174:
170:
166:
162:
158:
154:
150:
146:
142:
132:
130:
126:
122:
118:
114:
110:
105:
103:
99:
95:
91:
87:
83:
79:
75:
71:
67:
63:
59:
56:
55:combinatorial
49:
42:
38:
34:
30:
19:
7120:Applications
6953:Block design
6952:
6871:
6822:
6802:
6779:, Springer,
6776:
6758:
6752:
6720:
6714:
6689:
6683:
6664:
6663:. Series A.
6658:
6652:
6632:
6609:
6587:
6566:
6548:
6515:
6511:
6507:
6487:
6467:
6441:
6435:
6415:
6393:
6375:
6371:
6350:
6344:
6320:
6294:
6275:
6274:. Series A.
6269:
6213:
6209:
6199:
6187:
6178:
6166:
6154:
6142:
6130:
6118:
6106:
6094:
6082:
6070:
6058:
6051:Stinson 2003
6046:
6034:
6022:
6010:
6003:Stinson 2003
5998:
5986:
5975:
5964:
5951:
5944:
5932:
5927:, pp.320-335
5920:
5893:
5886:Stinson 2003
5881:
5873:
5869:
5865:
5861:
5857:
5853:
5849:
5844:
5816:
5805:
5794:
5781:
5754:
5748:
5738:
5731:Stinson 2003
5726:
5714:
5685:
5674:
5667:
5664:
5597:
5589:
5582:
5575:
5562:
5558:
5551:
5548:
5529:
5522:
5519:
5429:
5423:
5419:
5395:
5387:
5376:
5373:Applications
5342:
5340:
5324:
4996:
4994:
4984:
4980:
4978:
4753:
4463:
4456:
4449:
4442:
4426:
4422:
4418:
4414:
4412:
4375:
4373:
4336:
4299:
4262:
4225:
4188:
4151:
4115:
4111:
4110:if elements
4107:
4103:
4099:
4095:
4091:
4089:
4079:
4077:
4068:
4064:
4060:
4056:
4052:
4048:
4044:
4040:
4036:
4032:
4031:-set X with
4028:
4024:
4020:
4016:
4014:
4009:
4005:
4001:
3941:
3939:
3933:
3929:
3925:
3921:
3917:
3715:
3711:
3707:
3703:
3549:
3542:
3538:
3534:
3530:
3506:
3502:
3498:
3494:
3487:
3483:
3472:
3470:
3460:
3456:
3452:
3448:
3444:
3440:
3436:
3432:
3428:
3427:
3418:
3416:
3411:
3405:
3392:
3385:
3381:
3374:
3368:
3355:
3350:
3346:
3342:
3338:
3334:
3330:
3326:
3322:
3318:
3312:
3310:
3305:
3302:Möbius plane
3297:
3293:
3289:
3286:affine plane
3283:
3274:
3268:
3264:
3258:
3254:
3248:
3241:
3237:
3233:
3229:
3228:
3223:
3221:
3216:
3212:
3208:
3202:
3197:
3193:
3189:
3185:
3181:
3177:
3173:
3169:
3168:
3163:
3160:
3156:
3149:
3145:
3141:
3137:
3133:
3129:
3125:
3121:
3117:
3113:
3109:
3105:
3100:
3096:
3093:
3089:
3085:
3081:
3077:
3073:
3069:
3065:
3061:
3059:
3050:block design
3049:
3047:
3038:
3034:
3030:
3028:
3023:
3019:
3017:
3012:
3008:
3004:
3000:
2992:
2988:
2984:
2980:
2976:
2972:
2968:
2967:
2756:
2748:
2744:
2737:
2735:
2576:
2572:
2568:
2564:
2560:
2556:
2552:
2548:
2544:
2540:
2536:
2532:
2528:
2524:
2520:
2516:
2512:
2508:
2504:
2500:
2496:
2492:
2488:
2486:
2480:
2466:
2463:
2458:
2454:
2450:
2446:
2442:
2438:
2436:
2431:
2427:
2423:
2421:
2411:
2409:
2320:
2316:
2312:
2308:
2300:
2296:
2294:
2289:
2285:
2281:
2277:
2273:
2268:
2264:
2260:
2256:
2249:
2245:
2241:
2237:
2233:
2229:
2223:
2211:
2186:for details.
2181:
2175:
2171:
2139:
2132:
2121:
2117:
2104:
2099:
2095:
2091:
2087:
2083:
2079:
2075:
2071:
2067:
2063:
2059:
2057:
2048:
2043:
2039:
2035:
2027:
2025:
2020:
2016:
2012:
2008:
2003:
1999:
1995:
1991:
1987:
1983:
1979:
1977:
1972:
1968:
1964:
1960:
1956:
1952:
1948:
1944:
1940:
1936:
1932:
1930:
1868:
1864:
1859:
1845:
1836:
1834:
1765:
1760:
1756:
1752:
1748:
1744:
1740:
1732:
1728:
1724:
1720:
1716:
1714:
1709:
1707:
1400:
1392:
1383:
1034:
1030:
1023:
1019:
1013:
1003:
999:
995:
991:
987:
985:
976:
971:
967:
957:
952:
948:
944:
940:
936:
932:
928:
924:
920:
916:
912:
908:
904:
900:
896:
892:
888:
884:
880:
876:
874:
869:
865:
861:
857:
853:
851:
846:
842:
838:
834:
830:
826:
822:
818:
814:
810:
806:
802:
798:
794:
792:
725:
721:
717:
713:
711:
668:
664:
660:
656:
652:
648:
644:
640:
636:
632:
628:
624:
620:
616:
612:
610:
600:
594:
584:
574:
564:
557:
551:
540:
536:
532:
528:
524:
520:
516:
512:
510:
505:
501:
497:
493:
489:
485:
481:
477:
473:
469:
465:
461:
457:
453:
449:
445:
441:
437:
433:
429:
425:
423:
393:
356:
352:
348:
344:
336:
324:
320:
316:
314:
304:
300:
292:
286:
273:
271:
262:
258:
254:
253:) is called
250:
248:
243:
239:
237:
226:
224:
220:
219:-value. For
216:
212:
208:
204:
200:
196:
192:
188:
184:
180:
176:
172:
168:
164:
160:
156:
152:
148:
144:
140:
138:
128:
120:
116:
112:
109:block design
108:
106:
98:cryptography
73:
62:block design
61:
52:
6991:Statistics
6341:Bose, R. C.
5958:, p. 4
5402:block codes
5356:triangular;
5333:is a BIBD.
4999:) satisfy:
4106:) entry is
3942:commutative
3088:a point of
2133:complements
58:mathematics
7147:Categories
7020:Fano plane
6985:Hypergraph
6459:2440/15239
6258:References
5898:Ryser 1963
5858:projective
5721:, pp.17−19
5608:Treatment
5448:Treatment
5430:design.bib
5428:-function
5343:then known
4991:Properties
3791:such that
3535:associates
3164:extendable
3159:; we call
3072:) be a t-(
2757:Note that
2557:parameters
2432:resolution
2129:Fano plane
2032:Fano plane
1403:Fano plane
966:, is that
893:complement
533:parameters
414:Levi graph
293:non-binary
289:statistics
240:incomplete
175:, denoted
37:statistics
6970:Incidence
6873:MathWorld
6737:225335042
6706:125795689
6590:. Dover.
6540:120721016
6444:: 52–75,
6223:1203.5378
6194:, pg. 127
6173:, pg. 242
6137:, pg. 238
6125:, pg. 237
5925:Hall 1986
5874:symmetric
5773:0195-6698
5185:∑
5157:−
5136:λ
5105:∑
5080:−
5044:∑
4378:(3) are:
3859:∈
3814:∈
3776:∈
3743:∈
3684:∈
3638:∗
3625:Defining
3600:∈
3533:th–
3492:partition
3138:extension
3048:The term
2941:−
2930:−
2898:−
2887:−
2872:λ
2860:λ
2784:λ
2772:λ
2712:…
2674:−
2663:−
2632:−
2621:−
2603:λ
2591:λ
2551:, λ, and
2507:, called
2483:-designs)
2392:−
2366:−
2337:−
2174:(2,5) in
1905:−
1887:−
1809:−
1788:−
1779:λ
769:−
748:−
739:λ
464:, called
407:bipartite
404:biregular
229:) designs
147:) if all
121:2-design,
72:known as
7084:Theorems
6995:Blocking
6965:Geometry
6800:(1987).
6586:(1988).
6319:(1986),
5915:, pg.109
5690:See also
5617:Block C
5614:Block B
5611:Block A
4987:values.
4078:A PBIBD(
4075:blocks.
4000:for all
3521:, ..., R
3486:of size
3317:in PG(3,
3172:: If a
2555:are the
2495:-design
2437:If a 2-(
2407:blocks.
2255:, where
2228:of size
2120:= 4 and
2054:Biplanes
982:Examples
809:) where
720:) where
446:2-design
333:geometry
325:1-design
282:multiset
269:(PBDs).
141:balanced
135:Overview
129:t-design
78:symmetry
6532:2384014
6510:= 11".
6248:7586742
6228:Bibcode
6065:, pg.29
5939:, pg.55
5839:, p. 27
5592:= {1, 2
5585:= {1, 3
5578:= {2, 3
5432:of the
5362:cyclic;
4086:Example
3710:, then
3548:
3501:×
3475:-class
3431:. Let
3429:Theorem
3351:egglike
3230:Theorem
3170:Theorem
2969:Theorem
2303:is the
2272:is the
2240:matrix
2060:biplane
2034:, with
891:. The
305:regular
255:uniform
207:if the
181:regular
143:(up to
6941:Fields
6829:
6810:
6796:&
6783:
6735:
6704:
6639:
6616:
6594:
6573:
6555:
6538:
6530:
6496:
6474:
6424:
6402:
6332:
6301:
6246:
5866:square
5771:
5587:} and
5445:Block
5442:Plots
4128:
3443:); so
3232::. If
3092:. The
2971:: Any
2736:where
2509:blocks
2232:is an
2184:points
2146:after
1931:Since
1037:) is:
868:, and
667:, and
466:blocks
430:points
394:Every
339:, see
274:simple
259:proper
225:PBIBD(
167:. For
100:, and
74:blocks
64:is an
39:, see
31:. For
6733:S2CID
6702:S2CID
6536:S2CID
6244:S2CID
6218:arXiv
5956:(PDF)
5733:, p.1
5707:Notes
5681:λ = 1
5536:λ = 1
4019:with
3702:, if
3505:into
3414:). .
3314:ovoid
3300:, or
2754:= λ.
2111:digon
2004:lines
1751:is a
1743:is a
1737:Ryser
1389:2456.
994:= 3,
990:= 6,
877:order
539:>
511:Here
410:graph
7075:Dual
6827:ISBN
6808:ISBN
6781:ISBN
6655:= 9"
6637:ISBN
6614:ISBN
6592:ISBN
6571:ISBN
6553:ISBN
6494:ISBN
6472:ISBN
6422:ISBN
6400:ISBN
6330:ISBN
6299:ISBN
5769:ISSN
5566:and
5542:and
5534:and
5508:302
5497:301
5486:202
5475:201
5464:102
5453:101
5396:The
4983:and
4114:and
4090:Let
4059:are
4055:and
4008:and
3932:and
3836:and
3541:has
3509:+ 1
3060:Let
2845:and
2575:and
2491:, a
2263:and
2026:For
1761:X, B
943:′ =
931:′ =
919:′ =
911:′ =
903:′ =
875:The
856:and
813:and
655:and
484:and
450:BIBD
448:(or
161:pair
117:BIBD
60:, a
6763:doi
6725:doi
6694:doi
6669:doi
6520:doi
6454:hdl
6446:doi
6380:doi
6355:doi
6280:doi
6236:doi
5852:or
5759:doi
5670:= 2
5554:= 3
5532:= 2
5525:= 3
3944:if
3720:If
3714:in
3706:in
3517:, R
3513:, R
3497:of
3481:set
3471:An
3417:If
3226:).
3144:if
3140:of
3064:= (
3015:.)
2539:),
2515:in
2224:An
2176:PSL
2172:PSL
2102:).
2062:or
1955:= (
1943:as
1018:(a
837:in
500:in
488:in
472:in
331:in
323:or
287:In
278:set
257:or
53:In
35:in
7149::
6870:.
6855::
6757:,
6747:;
6731:.
6721:51
6719:.
6700:.
6690:48
6688:.
6665:24
6657:.
6631:,
6534:.
6528:MR
6526:.
6516:16
6514:.
6452:,
6442:10
6440:,
6376:47
6374:,
6351:20
6349:,
6323:,
6315:;
6276:11
6268:.
6242:.
6234:.
6226:.
6214:16
6212:.
5905:^
5827:^
5767:.
5755:28
5753:.
5747:.
5683:.
5672:.
5659:0
5656:1
5653:1
5650:3
5645:1
5642:0
5639:1
5636:2
5631:1
5628:1
5625:0
5622:1
5594:}.
5561:,
5527:,
5514:1
5511:3
5503:2
5500:3
5492:3
5489:2
5481:1
5478:2
5470:2
5467:1
5459:3
5456:1
5412:.
5393:.
5349::
4146:6
4122:.
4067:≤
4015:A
4004:,
3924:,
3920:,
3712:R*
3643::=
3425:.
3398:),
3391:,
3380:+
3308:.
3176:-(
3068:,
3045:.
3033:-(
3029:A
2995:,λ
2987:-(
2975:-(
2965:.
2563:-(
2547:,
2422:A
2414:.
2246:HH
2236:×
2216:.
2058:A
2019:=
1990:=
1982:=
1975:.
1947:=
1935:=
1727:=
1719:=
939:,
927:,
915:,
907:,
883:=
864:,
805:,
801:,
716:,
663:,
647:,
639:,
635:,
631:,
627:,
619:,
615:,
527:,
440:,
436:,
416:.
284:.
235:.
131:.
104:.
96:,
92:,
88:,
84:,
6905:e
6898:t
6891:v
6876:.
6835:.
6816:.
6765::
6759:9
6739:.
6727::
6708:.
6696::
6677:.
6671::
6653:k
6622:.
6600:.
6542:.
6522::
6508:k
6456::
6448::
6382::
6357::
6336:.
6288:.
6282::
6250:.
6238::
6230::
6220::
5876:.
5775:.
5761::
5677:C
5668:r
5590:C
5583:B
5576:A
5568:C
5563:B
5559:A
5552:b
5544:r
5540:b
5530:k
5523:v
5426:R
5331:2
5327:1
5308:j
5303:h
5300:i
5296:p
5290:j
5286:n
5282:=
5277:i
5272:h
5269:j
5265:p
5259:i
5255:n
5231:j
5227:n
5223:=
5218:h
5213:u
5210:j
5206:p
5200:m
5195:0
5192:=
5189:u
5163:)
5160:1
5154:k
5151:(
5148:r
5145:=
5140:i
5130:i
5126:n
5120:m
5115:1
5112:=
5109:i
5083:1
5077:v
5074:=
5069:i
5065:n
5059:m
5054:1
5051:=
5048:i
5022:k
5019:b
5016:=
5013:r
5010:v
4997:m
4985:r
4981:λ
4963:)
4957:4
4952:2
4947:2
4942:2
4937:1
4932:1
4925:2
4920:4
4915:2
4910:1
4905:2
4900:1
4893:2
4888:2
4883:4
4878:1
4873:1
4868:2
4861:2
4856:1
4851:1
4846:4
4841:2
4836:2
4829:1
4824:2
4819:1
4814:2
4809:4
4804:2
4797:1
4792:1
4787:2
4782:2
4777:2
4772:4
4766:(
4738:)
4732:1
4727:1
4722:1
4717:0
4712:1
4707:0
4702:0
4697:0
4690:1
4685:1
4680:0
4675:1
4670:0
4665:0
4660:1
4655:0
4648:1
4643:1
4638:0
4633:0
4628:0
4623:1
4618:0
4613:1
4606:0
4601:0
4596:1
4591:1
4586:1
4581:1
4576:0
4571:0
4564:0
4559:0
4554:1
4549:1
4544:0
4539:0
4534:1
4529:1
4522:0
4517:0
4512:0
4507:0
4502:1
4497:1
4492:1
4487:1
4481:(
4467:3
4464:n
4460:1
4457:n
4453:2
4450:n
4446:0
4443:n
4439:3
4435:2
4431:1
4427:r
4423:k
4419:b
4415:v
4376:A
4337:6
4300:5
4263:4
4226:3
4189:2
4152:1
4143:5
4140:4
4137:3
4134:2
4131:1
4120:s
4116:j
4112:i
4108:s
4104:j
4102:,
4100:i
4096:X
4092:A
4080:n
4073:i
4069:n
4065:i
4061:i
4057:y
4053:x
4049:X
4045:n
4041:r
4037:k
4033:b
4029:v
4025:n
4021:n
4010:k
4006:j
4002:i
3986:k
3981:i
3978:j
3974:p
3970:=
3965:k
3960:j
3957:i
3953:p
3936:.
3934:y
3930:x
3926:k
3922:j
3918:i
3902:k
3897:j
3894:i
3890:p
3867:j
3863:R
3856:)
3853:y
3850:,
3847:z
3844:(
3822:i
3818:R
3811:)
3808:z
3805:,
3802:x
3799:(
3779:X
3773:z
3751:k
3747:R
3740:)
3737:y
3734:,
3731:x
3728:(
3716:S
3708:S
3704:R
3690:}
3687:R
3681:)
3678:x
3675:,
3672:y
3669:(
3665:|
3661:)
3658:y
3655:,
3652:x
3649:(
3646:{
3634:R
3622:.
3606:}
3603:X
3597:x
3594::
3591:)
3588:x
3585:,
3582:x
3579:(
3576:{
3573:=
3568:0
3564:R
3550:i
3546:i
3543:n
3539:X
3531:i
3527:i
3523:n
3519:1
3515:0
3507:n
3503:X
3499:X
3495:S
3488:v
3484:X
3473:n
3461:q
3457:q
3453:q
3449:q
3445:q
3441:q
3437:q
3433:q
3419:q
3412:q
3396:4
3393:x
3389:3
3386:x
3384:(
3382:f
3378:2
3375:x
3372:1
3369:x
3347:q
3343:O
3339:q
3335:q
3331:O
3327:q
3323:q
3319:q
3306:n
3294:n
3290:n
3269:k
3265:v
3259:k
3255:v
3249:D
3242:k
3240:,
3238:v
3234:D
3217:n
3213:n
3209:n
3198:v
3196:(
3194:b
3190:k
3186:λ
3184:,
3182:k
3180:,
3178:v
3174:t
3161:D
3157:D
3153:p
3150:E
3146:E
3142:D
3134:E
3130:λ
3126:k
3122:v
3118:t
3114:D
3110:p
3106:X
3101:p
3097:D
3090:X
3086:p
3082:λ
3080:,
3078:k
3076:,
3074:v
3070:B
3066:X
3062:D
3039:k
3037:,
3035:v
3031:t
3024:t
3020:t
3013:s
3009:t
3005:s
3001:s
2997:s
2993:k
2991:,
2989:v
2985:s
2981:k
2979:,
2977:v
2973:t
2950:)
2944:1
2938:t
2933:1
2927:k
2921:(
2914:/
2907:)
2901:1
2895:t
2890:1
2884:v
2878:(
2869:=
2864:1
2856:=
2853:r
2830:)
2825:t
2822:k
2817:(
2810:/
2803:)
2798:t
2795:v
2790:(
2781:=
2776:0
2768:=
2765:b
2751:t
2749:λ
2745:i
2740:i
2738:λ
2721:,
2718:t
2715:,
2709:,
2706:1
2703:,
2700:0
2697:=
2694:i
2683:)
2677:i
2671:t
2666:i
2660:k
2654:(
2647:/
2641:)
2635:i
2629:t
2624:i
2618:v
2612:(
2600:=
2595:i
2577:r
2573:b
2569:k
2567:,
2565:v
2561:t
2553:t
2549:r
2545:k
2541:b
2537:X
2533:v
2529:T
2525:t
2521:r
2517:X
2513:x
2505:X
2501:k
2497:B
2493:t
2489:t
2481:t
2459:c
2455:v
2451:b
2447:c
2443:k
2441:,
2439:v
2412:a
2395:1
2389:a
2369:1
2363:a
2360:2
2340:1
2334:a
2331:4
2317:a
2313:a
2309:a
2301:M
2297:a
2290:m
2286:m
2278:m
2274:m
2269:m
2265:I
2261:H
2257:H
2253:m
2250:I
2242:H
2238:m
2234:m
2230:m
2199:.
2182:p
2162:.
2122:k
2118:v
2113:.
2100:k
2096:r
2092:n
2088:n
2084:v
2080:n
2076:k
2072:n
2068:λ
2044:n
2040:n
2036:v
2028:n
2021:n
2017:r
2013:n
2009:k
2000:b
1996:n
1992:n
1988:b
1984:v
1980:b
1973:n
1969:n
1965:n
1961:n
1957:n
1953:v
1949:k
1945:n
1937:r
1933:k
1914:.
1911:)
1908:1
1902:k
1899:(
1896:k
1893:=
1890:1
1884:v
1869:n
1865:λ
1837:v
1818:.
1815:)
1812:1
1806:k
1803:(
1800:k
1797:=
1794:)
1791:1
1785:v
1782:(
1757:k
1753:v
1749:B
1745:v
1741:X
1733:λ
1729:v
1725:b
1721:k
1717:r
1687:)
1680:0
1675:1
1670:1
1665:0
1660:1
1655:0
1650:0
1643:1
1638:0
1633:0
1628:1
1623:1
1618:0
1613:0
1606:1
1601:0
1596:1
1591:0
1586:0
1581:1
1576:0
1569:0
1564:1
1559:0
1554:1
1549:0
1544:1
1539:0
1532:1
1527:1
1522:0
1517:0
1512:0
1507:0
1502:1
1495:0
1490:0
1485:1
1480:1
1475:0
1470:0
1465:1
1458:0
1453:0
1448:0
1443:0
1438:1
1433:1
1428:1
1421:(
1367:)
1361:1
1356:0
1351:1
1346:0
1341:1
1336:1
1331:1
1326:0
1321:0
1316:0
1309:0
1304:1
1299:1
1294:1
1289:0
1284:1
1279:0
1274:1
1269:0
1264:0
1257:1
1252:1
1247:0
1242:1
1237:0
1232:0
1227:1
1222:0
1217:1
1212:0
1205:1
1200:1
1195:0
1190:0
1185:1
1180:0
1175:0
1170:1
1165:0
1160:1
1153:0
1148:0
1143:1
1138:1
1133:1
1128:0
1123:0
1118:0
1113:1
1108:1
1101:0
1096:0
1091:0
1086:0
1081:0
1076:1
1071:1
1066:1
1061:1
1056:1
1050:(
1035:k
1031:r
1024:b
1022:×
1020:v
1004:r
1000:b
996:λ
992:k
988:v
972:v
968:b
953:r
949:b
945:λ
941:λ
937:k
933:v
929:k
925:r
921:b
917:r
913:b
909:b
905:v
901:v
897:X
889:λ
885:r
881:n
870:λ
866:k
862:v
858:r
854:b
847:r
843:r
839:X
835:x
831:x
827:r
823:x
819:B
815:y
811:x
807:B
803:y
799:x
795:x
778:,
775:)
772:1
766:k
763:(
760:r
757:=
754:)
751:1
745:v
742:(
726:p
722:B
718:p
714:B
697:,
694:r
691:v
688:=
685:k
682:b
669:λ
665:k
661:v
657:r
653:b
649:k
645:v
641:λ
637:k
633:r
629:b
625:v
621:λ
617:k
613:v
601:t
595:λ
585:k
575:r
565:b
558:X
552:v
541:k
537:v
529:r
525:k
521:b
517:X
513:v
506:r
502:X
498:x
494:λ
490:X
486:y
482:x
478:r
474:X
470:x
462:X
458:k
454:B
442:λ
438:r
434:k
426:X
379:r
376:v
373:=
370:k
367:b
357:b
353:v
349:r
345:k
317:t
263:t
251:k
244:k
227:n
221:t
217:λ
213:n
209:t
201:t
197:t
193:λ
189:t
185:t
177:r
169:t
157:t
153:λ
149:t
145:t
115:(
50:.
43:.
20:)
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