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250px, thumbnail, The Fano plane is a Steiner triple system S(2,3,7). The blocks are the 7 lines, each containing 3 points. Every pair of points belongs to a unique line. In
combinatorial Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ap ...
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a Steiner system (named after
Jakob Steiner Jakob Steiner (18 March 1796 – 1 April 1863) was a Swiss mathematician who worked primarily in geometry. Life Steiner was born in the village of Utzenstorf, Canton of Bern. At 18, he became a pupil of Heinrich Pestalozzi and afterwards st ...
) is a type of
block design In combinatorial mathematics, a block design is an incidence structure consisting of a set together with a family of subsets known as ''blocks'', chosen such that frequency of the elements satisfies certain conditions making the collection of bl ...
, specifically a t-design with λ = 1 and ''t'' = 2 or (recently) ''t'' ≥ 2. A Steiner system with parameters ''t'', ''k'', ''n'', written S(''t'',''k'',''n''), is an ''n''-element
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
''S'' together with a set of ''k''-element
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
s of ''S'' (called blocks) with the property that each ''t''-element subset of ''S'' is contained in exactly one block. In an alternate notation for block designs, an S(''t'',''k'',''n'') would be a ''t''-(''n'',''k'',1) design. This definition is relatively new. The classical definition of Steiner systems also required that ''k'' = ''t'' + 1. An S(2,3,''n'') was (and still is) called a ''Steiner triple'' (or ''triad'') ''system'', while an S(3,4,''n'') is called a ''Steiner quadruple system'', and so on. With the generalization of the definition, this naming system is no longer strictly adhered to. Long-standing problems in
design theory Design theory is a subfield of design research concerned with various theoretical approaches towards understanding and delineating design principles, design knowledge, and design practice. History Design theory has been approached and interp ...
were whether there exist any nontrivial Steiner systems (nontrivial meaning ''t'' < ''k'' < ''n'') with ''t'' ≥ 6; also whether infinitely many have ''t'' = 4 or 5. Both existences were proved by
Peter Keevash Peter Keevash (born 30 November 1978) is a British mathematician, working in combinatorics. He is Professor of Mathematics at the University of Oxford and a Fellow of Mansfield College. Early years Keevash was born in Brighton, England, but ...
in 2014. His proof is non-constructive and, as of 2019, no actual Steiner systems are known for large values of ''t''.


Types of Steiner systems

A finite
projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that d ...
of order , with the lines as blocks, is an , since it has points, each line passes through points, and each pair of distinct points lies on exactly one line. A finite affine plane of order , with the lines as blocks, is an . An affine plane of order can be obtained from a projective plane of the same order by removing one block and all of the points in that block from the projective plane. Choosing different blocks to remove in this way can lead to non-isomorphic affine planes. An S(3,4,''n'') is called a Steiner quadruple system. A necessary and sufficient condition for the existence of an S(3,4,''n'') is that ''n'' \equiv 2 or 4 (mod 6). The abbreviation SQS(''n'') is often used for these systems. Up to isomorphism, SQS(8) and SQS(10) are unique, there are 4 SQS(14)s and 1,054,163 SQS(16)s. An S(4,5,''n'') is called a Steiner quintuple system. A necessary condition for the existence of such a system is that ''n'' \equiv 3 or 5 (mod 6) which comes from considerations that apply to all the classical Steiner systems. An additional necessary condition is that ''n'' \not\equiv 4 (mod 5), which comes from the fact that the number of blocks must be an integer. Sufficient conditions are not known. There is a unique Steiner quintuple system of order 11, but none of order 15 or order 17. Systems are known for orders 23, 35, 47, 71, 83, 107, 131, 167 and 243. The smallest order for which the existence is not known (as of 2011) is 21.


Steiner triple systems

An S(2,3,''n'') is called a Steiner triple system, and its blocks are called triples. It is common to see the abbreviation STS(''n'') for a Steiner triple system of order ''n''. The total number of pairs is ''n(n-1)/2'', of which three appear in a triple, and so the total number of triples is ''n''(''n''−1)/6. This shows that ''n'' must be of the form ''6k+1'' or ''6k + 3'' for some ''k''. The fact that this condition on ''n'' is sufficient for the existence of an S(2,3,''n'') was proved by
Raj Chandra Bose Raj Chandra Bose (19 June 1901 – 31 October 1987) was an Indian American mathematician and statistician best known for his work in design theory, finite geometry and the theory of error-correcting codes in which the class of BCH codes is pa ...
and T. Skolem. The projective plane of order 2 (the
Fano plane In finite geometry, the Fano plane (after Gino Fano) is a finite projective plane with the smallest possible number of points and lines: 7 points and 7 lines, with 3 points on every line and 3 lines through every point. These points and lines ...
) is an STS(7) and the affine plane of order 3 is an STS(9). Up to isomorphism, the STS(7) and STS(9) are unique, there are two STS(13)s, 80 STS(15)s, and 11,084,874,829 STS(19)s. We can define a multiplication on the set ''S'' using the Steiner triple system by setting ''aa'' = ''a'' for all ''a'' in ''S'', and ''ab'' = ''c'' if is a triple. This makes ''S'' an
idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
,
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
quasigroup. It has the additional property that ''ab'' = ''c'' implies ''bc'' = ''a'' and ''ca'' = ''b''. Conversely, any (finite) quasigroup with these properties arises from a Steiner triple system. Commutative idempotent quasigroups satisfying this additional property are called ''Steiner quasigroups''.


Resolvable Steiner systems

Some of the S(2,3,n) systems can have their blocks be partitioned into (n-1)/2 sets of (n/3) triples each. This is called ''resolvable'' and such systems are called ''Kirkman triple systems'' after Thomas Kirkman, who studied such resolvable systems before Steiner. Dale Mesner, Earl Kramer, and others investigated collections of Steiner triple systems that are mutually disjoint (i.e., no two Steiner systems in such a collection share a common triplet). It is known (Bays 1917, Kramer & Mesner 1974) that seven different S(2,3,9) systems can be generated to together cover all 84 triplets on a 9-set; it was also known by them that there are 15360 different ways to find such 7-sets of solutions, which reduce to two non-isomorphic solutions under relabeling, with multiplicities 6720 and 8640 respectively. The corresponding question for finding thirteen different disjoint S(2,3,15) systems was asked by James Sylvester in 1860 as an extension of the
Kirkman's schoolgirl problem Kirkman's schoolgirl problem is a problem in combinatorics proposed by Rev. Thomas Penyngton Kirkman in 1850 as Query VI in '' The Lady's and Gentleman's Diary'' (pg.48). The problem states: Fifteen young ladies in a school walk out three abre ...
, namely whether Kirkman's schoolgirls could march for an entire term of 13 weeks with no triplet of girls being repeated over the whole term. The question was solved by RHF Denniston in 1974, who constructed Week 1 as follows:
Day 1 ABJ CEM FKL HIN DGO
Day 2 ACH DEI FGM JLN BKO
Day 3 ADL BHM GIK CFN EJO
Day 4 AEG BIL CJK DMN FHO
Day 5 AFI BCD GHJ EKN LMO
Day 6 AKM DFJ EHL BGN CIO
Day 7 BEF CGL DHK IJM ANO
for girls labeled A to O, and constructed each subsequent week's solution from its immediate predecessor by changing A to B, B to C, ... L to M and M back to A, all while leaving N and O unchanged. The Week 13 solution, upon undergoing that relabeling, returns to the Week 1 solution. Denniston reported in his paper that the search he employed took 7 hours on an Elliott 4130 computer at the
University of Leicester , mottoeng = So that they may have life , established = , type = public research university , endowment = £20.0 million , budget = £326 million , chancellor = David Willetts , vice_chancellor = Nishan Canagarajah , head_lab ...
, and he immediately ended the search on finding the solution above, not looking to establish uniqueness. The number of non-isomorphic solutions to Sylvester's problem remains unknown as of 2021.


Properties

It is clear from the definition of that 1 < t < k < n. (Equalities, while technically possible, lead to trivial systems.) If exists, then taking all blocks containing a specific element and discarding that element gives a ''derived system'' . Therefore, the existence of is a necessary condition for the existence of . The number of -element subsets in is \tbinom n t, while the number of -element subsets in each block is \tbinom k t. Since every -element subset is contained in exactly one block, we have \tbinom n t = b\tbinom k t, or :b = \frac = \frac, where is the number of blocks. Similar reasoning about -element subsets containing a particular element gives us \tbinom=r\tbinom, or :r=\frac =\frac, where is the number of blocks containing any given element. From these definitions follows the equation bk=rn. It is a necessary condition for the existence of that and are integers. As with any block design, Fisher's inequality b\ge n is true in Steiner systems. Given the parameters of a Steiner system and a subset of size t' \leq t, contained in at least one block, one can compute the number of blocks intersecting that subset in a fixed number of elements by constructing a Pascal triangle. In particular, the number of blocks intersecting a fixed block in any number of elements is independent of the chosen block. The number of blocks that contain any ''i''-element set of points is: : \lambda_i = \left.\binom \right/ \binom \text i = 0,1,\ldots,t, It can be shown that if there is a Steiner system , where is a prime power greater than 1, then \equiv 1 or . In particular, a Steiner triple system must have . And as we have already mentioned, this is the only restriction on Steiner triple systems, that is, for each
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
, systems and exist.


History

Steiner triple systems were defined for the first time by Wesley S. B. Woolhouse in 1844 in the Prize question #1733 of Lady's and Gentlemen's Diary. The posed problem was solved by . In 1850 Kirkman posed a variation of the problem known as
Kirkman's schoolgirl problem Kirkman's schoolgirl problem is a problem in combinatorics proposed by Rev. Thomas Penyngton Kirkman in 1850 as Query VI in '' The Lady's and Gentleman's Diary'' (pg.48). The problem states: Fifteen young ladies in a school walk out three abre ...
, which asks for triple systems having an additional property (resolvability). Unaware of Kirkman's work, reintroduced triple systems, and as this work was more widely known, the systems were named in his honor.


Mathieu groups

Several examples of Steiner systems are closely related to
group theory In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...
. In particular, the
finite simple groups Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb Traditionally, a finite verb (from la, fīnītus, past particip ...
called
Mathieu group In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups ''M''11, ''M''12, ''M''22, ''M''23 and ''M''24 introduced by . They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 obje ...
s arise as
automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
s of Steiner systems: * The Mathieu group M11 is the automorphism group of a S(4,5,11) Steiner system * The Mathieu group M12 is the automorphism group of a S(5,6,12) Steiner system * The Mathieu group M22 is the unique index 2 subgroup of the automorphism group of a S(3,6,22) Steiner system * The Mathieu group M23 is the automorphism group of a S(4,7,23) Steiner system * The Mathieu group M24 is the automorphism group of a S(5,8,24) Steiner system.


The Steiner system S(5, 6, 12)

There is a unique S(5,6,12) Steiner system; its automorphism group is the
Mathieu group In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups ''M''11, ''M''12, ''M''22, ''M''23 and ''M''24 introduced by . They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 obje ...
M12, and in that context it is denoted by W12.


Projective line construction

This construction is due to Carmichael (1937). Add a new element, call it , to the 11 elements of the
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
11 (that is, the integers mod 11). This set, , of 12 elements can be formally identified with the points of the
projective line In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a ''point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; ...
over 11. Call the following specific subset of size 6, :\, a "block" (it contains together with the 5 nonzero squares in 11). From this block, we obtain the other blocks of the (5,6,12) system by repeatedly applying the
linear fractional transformation In mathematics, a linear fractional transformation is, roughly speaking, a transformation of the form :z \mapsto \frac , which has an inverse. The precise definition depends on the nature of , and . In other words, a linear fractional transfo ...
s: :z' = f(z) = \frac, where are in 11 and . With the usual conventions of defining and , these functions map the set onto itself. In geometric language, they are projectivities of the projective line. They form a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
under composition which is the
projective special linear group In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associa ...
(2,11) of order 660. There are exactly five elements of this group that leave the starting block fixed setwise, namely those such that and so that . So there will be 660/5 = 132 images of that block. As a consequence of the multiply transitive property of this group
acting Acting is an activity in which a story is told by means of its enactment by an actor or actress who adopts a character—in theatre, television, film, radio, or any other medium that makes use of the mimetic mode. Acting involves a broad r ...
on this set, any subset of five elements of will appear in exactly one of these 132 images of size six.


Kitten construction

An alternative construction of W12 is obtained by use of the 'kitten' of R.T. Curtis, which was intended as a "hand calculator" to write down blocks one at a time. The kitten method is based on completing patterns in a 3x3 grid of numbers, which represent an
affine geometry In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting") the metric notions of distance and angle. As the notion of '' parallel lines'' is one of the main properties that is ...
on the
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
F3xF3, an S(2,3,9) system.


Construction from K6 graph factorization

The relations between the graph factors of the complete graph K6 generate an S(5,6,12). A K6 graph has 6 vertices, 15 edges, 15
perfect matching In graph theory, a perfect matching in a graph is a matching that covers every vertex of the graph. More formally, given a graph , a perfect matching in is a subset of edge set , such that every vertex in the vertex set is adjacent to exactl ...
s, and 6 different 1-factorizations (ways to partition the edges into disjoint perfect matchings). The set of vertices (labeled 123456) and the set of factorizations (labeled ''ABCDEF'') provide one block each. Every pair of factorizations has exactly one perfect matching in common. Suppose factorizations ''A'' and ''B'' have the common matching with edges 12, 34 and 56. Add three new blocks ''AB''3456, 12''AB''56, and 1234''AB'', replacing each edge in the common matching with the factorization labels in turn. Similarly add three more blocks 12''CDEF'', 34''CDEF'', and 56''CDEF'', replacing the factorization labels by the corresponding edge labels of the common matching. Do this for all 15 pairs of factorizations to add 90 new blocks. Finally, take the full set of \tbinom = 924 combinations of 6 objects out of 12, and discard any combination that has 5 or more objects in common with any of the 92 blocks generated so far. Exactly 40 blocks remain, resulting in blocks of the S(5,6,12). This method works because there is an outer automorphism on the symmetric group ''S''6, which maps the vertices to factorizations and the edges to partitions. Permuting the vertices causes the factorizations to permute differently, in accordance with the outer automorphism.


The Steiner system S(5, 8, 24)

The Steiner system S(5, 8, 24), also known as the Witt design or Witt geometry, was first described by and rediscovered by . This system is connected with many of the
sporadic simple group In mathematics, a sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups. A simple group is a group ''G'' that does not have any normal subgroups except for the trivial group and ''G'' itself. T ...
s and with the exceptional 24-dimensional
lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an orna ...
known as the Leech lattice. The automorphism group of S(5, 8, 24) is the Mathieu group M24, and in that context the design is denoted W24 ("W" for "Witt")


Direct lexicographic generation

All 8-element subsets of a 24-element set are generated in lexicographic order, and any such subset which differs from some subset already found in fewer than four positions is discarded. The list of octads for the elements 01, 02, 03, ..., 22, 23, 24 is then: :: 01 02 03 04 05 06 07 08 :: 01 02 03 04 09 10 11 12 :: 01 02 03 04 13 14 15 16 :: . :: . (next 753 octads omitted) :: . :: 13 14 15 16 17 18 19 20 :: 13 14 15 16 21 22 23 24 :: 17 18 19 20 21 22 23 24 Each single element occurs 253 times somewhere in some octad. Each pair occurs 77 times. Each triple occurs 21 times. Each quadruple (tetrad) occurs 5 times. Each quintuple (pentad) occurs once. Not every hexad, heptad or octad occurs.


Construction from the binary Golay code

The 4096 codewords of the 24-bit
binary Golay code In mathematics and electronics engineering, a binary Golay code is a type of linear error-correcting code used in digital communications. The binary Golay code, along with the ternary Golay code, has a particularly deep and interesting connection ...
are generated, and the 759 codewords with a
Hamming weight The Hamming weight of a string is the number of symbols that are different from the zero-symbol of the alphabet used. It is thus equivalent to the Hamming distance from the all-zero string of the same length. For the most typical case, a string ...
of 8 correspond to the S(5,8,24) system. The Golay code can be constructed by many methods, such as generating all 24-bit binary strings in lexicographic order and discarding those that differ from some earlier one in fewer than 8 positions. The result looks like this:
    000000000000000000000000
    000000000000000011111111
    000000000000111100001111
    .
    . (next 4090 24-bit strings omitted)
    .
    111111111111000011110000
    111111111111111100000000
    111111111111111111111111
The codewords form a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
under the
XOR Exclusive or or exclusive disjunction is a logical operation that is true if and only if its arguments differ (one is true, the other is false). It is symbolized by the prefix operator J and by the infix operators XOR ( or ), EOR, EXOR, , ...
operation.


Projective line construction

This construction is due to Carmichael (1931). Add a new element, call it , to the 23 elements of the
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
23 (that is, the integers mod 23). This set, , of 24 elements can be formally identified with the points of the
projective line In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a ''point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; ...
over 23. Call the following specific subset of size 8, :\, a "block". (We can take any octad of the extended
binary Golay code In mathematics and electronics engineering, a binary Golay code is a type of linear error-correcting code used in digital communications. The binary Golay code, along with the ternary Golay code, has a particularly deep and interesting connection ...
, seen as a quadratic residue code.) From this block, we obtain the other blocks of the (5,8,24) system by repeatedly applying the
linear fractional transformation In mathematics, a linear fractional transformation is, roughly speaking, a transformation of the form :z \mapsto \frac , which has an inverse. The precise definition depends on the nature of , and . In other words, a linear fractional transfo ...
s: :z' = f(z) = \frac, where are in 23 and . With the usual conventions of defining and , these functions map the set onto itself. In geometric language, they are projectivities of the projective line. They form a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
under composition which is the
projective special linear group In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associa ...
(2,23) of order 6072. There are exactly 8 elements of this group that leave the initial block fixed setwise. So there will be 6072/8 = 759 images of that block. These form the octads of S(5,8,24).


Construction from the Miracle Octad Generator

The Miracle Octad Generator (MOG) is a tool to generate octads, such as those containing specified subsets. It consists of a 4x6 array with certain weights assigned to the rows. In particular, an 8-subset should obey three rules in order to be an octad of S(5,8,24). First, each of the 6 columns should have the same
parity Parity may refer to: * Parity (computing) ** Parity bit in computing, sets the parity of data for the purpose of error detection ** Parity flag in computing, indicates if the number of set bits is odd or even in the binary representation of the ...
, that is, they should all have an odd number of cells or they should all have an even number of cells. Second, the top row should have the same parity as each of the columns. Third, the rows are respectively multiplied by the weights 0, 1, 2, and 3 over the finite field of order 4, and column sums are calculated for the 6 columns, with multiplication and addition using the finite field arithmetic definitions. The resulting column sums should form a valid '' hexacodeword'' of the form where ''a, b, c'' are also from the finite field of order 4. If the column sums' parities don't match the row sum parity, or each other, or if there do not exist ''a, b, c'' such that the column sums form a valid hexacodeword, then that subset of 8 is not an octad of S(5,8,24). The MOG is based on creating a
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
(Conwell 1910, "The three-space PG(3,2) and its group") between the 35 ways to partition an 8-set into two different 4-sets, and the 35 lines of the Fano 3-space PG(3,2). It is also geometrically related (Cullinane, "Symmetry Invariance in a Diamond Ring", Notices of the AMS, pp A193-194, Feb 1979) to the 35 different ways to partition a 4x4 array into 4 different groups of 4 cells each, such that if the 4x4 array represents a four-dimensional finite
affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...
, then the groups form a set of parallel subspaces.


See also

* Constant weight code *
Kirkman's schoolgirl problem Kirkman's schoolgirl problem is a problem in combinatorics proposed by Rev. Thomas Penyngton Kirkman in 1850 as Query VI in '' The Lady's and Gentleman's Diary'' (pg.48). The problem states: Fifteen young ladies in a school walk out three abre ...
* Sylvester–Gallai configuration


Notes


References


References

* . * * . 2nd ed. (1999) . * * * * * * . * * * * . *


External links

* *
Steiner systems
by Andries E. Brouwer

by Dr. Alberto Delgado, Gabe Hart, and Michael Kolkebeck

by Johan E. Mebius {{Incidence structures Combinatorial design Design of experiments Families of sets de:Steiner-Tripel-System