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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Combinatorial design | 4/5 | https://en.wikipedia.org/wiki/Combinatorial_design | reference | science, encyclopedia | 2026-05-05T09:49:37.423591+00:00 | kb-cron |
An H(2n − 1, 2n) is a Room square of side 2n − 1, and thus the Howell designs generalize the concept of Room squares. The pairs of symbols in the cells of a Howell design can be thought of as the edges of an s regular graph on 2n vertices, called the underlying graph of the Howell design. Cyclic Howell designs are used as Howell movements in duplicate bridge tournaments. The rows of the design represent the rounds, the columns represent the boards, and the diagonals represent the tables. Linear spaces An (n,k,p,t)-lotto design is an n-set V of elements together with a set β of k-element subsets of V (blocks), so that for any p-subset P of V, there is a block B in β for which |P ∩ B | ≥ t. L(n,k,p,t) denotes the smallest number of blocks in any (n,k,p,t)-lotto design. The following is a (7,5,4,3)-lotto design with the smallest possible number of blocks: {1,2,3,4,7} {1,2,5,6,7} {3,4,5,6,7}. Lotto designs model any lottery that is run in the following way: Individuals purchase tickets consisting of k numbers chosen from a set of n numbers. At a certain point the sale of tickets is stopped and a set of p numbers is randomly selected from the n numbers. These are the winning numbers. If any sold ticket contains t or more of the winning numbers, a prize is given to the ticket holder. Larger prizes go to tickets with more matches. The value of L(n,k,p,t) is of interest to both gamblers and researchers, as this is the smallest number of tickets that are needed to be purchased in order to guarantee a prize. The Hungarian Lottery is a (90,5,5,t)-lotto design and it is known that L(90,5,5,2) = 100. Lotteries with parameters (49,6,6,t) are also popular worldwide and it is known that L(49,6,6,2) = 19. In general though, these numbers are hard to calculate and remain unknown. A geometric construction of one such design is given in Transylvanian lottery. Magic squares A (v,k,λ)-Mendelsohn design, or MD(v,k,λ), is a v-set V and a collection β of ordered k-tuples of distinct elements of V (called blocks), such that each ordered pair (x,y) with x ≠ y of elements of V is cyclically adjacent in λ blocks. The ordered pair (x,y) of distinct elements is cyclically adjacent in a block if the elements appear in the block as (...,x,y,...) or (y,...,x). An MD(v,3,λ) is a Mendelsohn triple system, MTS(v,λ). An example of an MTS(4,1) on V = {0,1,2,3} is: (0,1,2) (1,0,3) (2,1,3) (0,2,3) Any triple system can be made into a Mendelson triple system by replacing the unordered triple {a,b,c} with the pair of ordered triples (a,b,c) and (a,c,b), but as the example shows, the converse of this statement is not true. If (Q,∗) is an idempotent semisymmetric quasigroup, that is, x ∗ x = x (idempotent) and x ∗ (y ∗ x) = y (semisymmetric) for all x, y in Q, let β = {(x,y,x ∗ y): x, y in Q}. Then (Q, β) is a Mendelsohn triple system MTS(|Q|,1). This construction is reversible. Orthogonal arrays Packing designs A quasi-3 design is a symmetric design (SBIBD) in which each triple of blocks intersect in either x or y points, for fixed x and y called the triple intersection numbers (x < y). Any symmetric design with λ ≤ 2 is a quasi-3 design with x = 0 and y = 1. The point-hyperplane design of PG(n,q) is a quasi-3 design with x = (qn−2 − 1)/(q − 1) and y = λ = (qn−1 − 1)/(q − 1). If y = λ for a quasi-3 design, the design is isomorphic to PG(n,q) or a projective plane. A t-(v,k,λ) design D is quasi-symmetric with intersection numbers x and y (x < y) if every two distinct blocks intersect in either x or y points. These designs naturally arise in the investigation of the duals of designs with λ = 1. A non-symmetric (b > v) 2-(v,k,1) design is quasisymmetric with x = 0 and y = 1. A multiple (repeat all blocks a certain number of times) of a symmetric 2-(v,k,λ) design is quasisymmetric with x = λ and y = k. Hadamard 3-designs (extensions of Hadamard 2-designs) are quasisymmetric. Every quasisymmetric block design gives rise to a strongly regular graph (as its block graph), but not all SRGs arise in this way. The incidence matrix of a quasisymmetric 2-(v,k,λ) design with k ≡ x ≡ y (mod 2) generates a binary self-orthogonal code (when bordered if k is odd). Room squares A spherical design is a finite set X of points in a (d − 1)-dimensional sphere such that, for some integer t, the average value on X of every polynomial
f
(
x
1
,
…
,
x
d
)
{\displaystyle f(x_{1},\ldots ,x_{d})\ }
of total degree at most t is equal to the average value of f on the whole sphere, i.e., the integral of f divided by the area of the sphere. Turán systems An r × n tuscan-k rectangle on n symbols has r rows and n columns such that: each row is a permutation of the n symbols and for any two distinct symbols a and b and for each m from 1 to k, there is at most one row in which b is m steps to the right of a. If r = n and k = 1 these are referred to as Tuscan squares, while if r = n and k = n − 1 they are Florentine squares. A Roman square is a Tuscan square which is also a latin square (these are also known as row complete Latin squares). A Vatican square is a Florentine square which is also a Latin square. The following example is a tuscan-1 square on 7 symbols which is not tuscan-2: