Steiner system
A Steiner system with parameters t, k, n, written S(t,k,n), is an n-element set S together with a set of k-element subsets of S (called blocks) with the property that each t-element subset of S is contained in exactly one block.
With the generalization of the definition, this naming system is no longer strictly adhered to.
Choosing different blocks to remove in this way can lead to non-isomorphic affine planes.
3 or 5 (mod 6) which comes from considerations that apply to all the classical Steiner systems.
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[7] and T.
[note 1] Conversely, any (finite) quasigroup with these properties arises from a Steiner triple system.
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, 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,[11] who constructed Week 1 as follows: 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.
Denniston reported in his paper that the search he employed took 7 hours on an Elliott 4130 computer at the University of Leicester, 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.
The number of blocks that contain any i-element set of points is: It can be shown that if there is a Steiner system S(2, k, n), where k is a prime power greater than 1, then n
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.
In 1910 Geoffrey Thomas Bennett gave a graphical representation for Steiner triple systems.
[14][15][16] Several examples of Steiner systems are closely related to group theory.
This set, S, of 12 elements can be formally identified with the points of the projective line over F11.
Call the following specific subset of size 6, a "block" (it contains ∞ together with the 5 nonzero squares in F11).
With the usual conventions of defining f (−d/c) = ∞ and f (∞) = a/c, these functions map the set S onto itself.
There are exactly five elements of this group that leave the starting block fixed setwise,[18] namely those such that b=c=0 and ad=1 so that f(z) = a2 z.
As a consequence of the multiply transitive property of this group acting on this set, any subset of five elements of S will appear in exactly one of these 132 images of size six.
An alternative construction of W12 is obtained by use of the 'kitten' of R.T. Curtis,[19] 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 on the vector space F3xF3, an S(2,3,9) system.
Add three new blocks AB3456, 12AB56, and 1234AB, replacing each edge in the common matching with the factorization labels in turn.
This method works because there is an outer automorphism on the symmetric group S6, which maps the vertices to factorizations and the edges to partitions.
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.
This set, S, of 24 elements can be formally identified with the points of the projective line over F23.
There are exactly 8 elements of this group that leave the initial block fixed setwise.
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.
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, then the groups form a set of parallel subspaces.
