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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Combinatorial design | 3/5 | https://en.wikipedia.org/wiki/Combinatorial_design | reference | science, encyclopedia | 2026-05-05T09:49:37.423591+00:00 | kb-cron |
== Other combinatorial designs == The Handbook of Combinatorial Designs (Colbourn & Dinitz 2007) has, amongst others, 65 chapters, each devoted to a combinatorial design other than those given above. A partial listing is given below:
Association schemes A balanced ternary design BTD(V, B; ρ1, ρ2, R; K, Λ) is an arrangement of V elements into B multisets (blocks), each of cardinality K (K ≤ V), satisfying: Each element appears R = ρ1 + 2ρ2 times altogether, with multiplicity one in exactly ρ1 blocks and multiplicity two in exactly ρ2 blocks. Every pair of distinct elements appears Λ times (counted with multiplicity); that is, if mvb is the multiplicity of the element v in block b, then for every pair of distinct elements v and w,
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{\displaystyle \sum _{b=1}^{B}m_{vb}m_{wb}=\Lambda }
. For example, one of the only two nonisomorphic BTD(4,8;2,3,8;4,6)s (blocks are columns) is:
The incidence matrix of a BTD (where the entries are the multiplicities of the elements in the blocks) can be used to form a ternary error-correcting code analogous to the way binary codes are formed from the incidence matrices of BIBDs. A balanced tournament design of order n (a BTD(n)) is an arrangement of all the distinct unordered pairs of a 2n-set V into an n × (2n − 1) array such that every element of V appears precisely once in each column, and every element of V appears at most twice in each row. An example of a BTD(3) is given by
The columns of a BTD(n) provide a 1-factorization of the complete graph on 2n vertices, K2n. BTD(n)s can be used to schedule round-robin tournaments: the rows represent the locations, the columns the rounds of play and the entries are the competing players or teams. Bent functions Costas arrays Covering designs Factorial designs A frequency square (F-square) is a higher order generalization of a Latin square. Let S = {s1,s2, ..., sm} be a set of distinct symbols and (λ1, λ2, ...,λm) a frequency vector of positive integers. A frequency square of order n is an n × n array in which each symbol si occurs λi times, i = 1,2,...,m, in each row and column. The order n = λ1 + λ2 + ... + λm. An F-square is in standard form if in the first row and column, all occurrences of si precede those of sj whenever i < j. A frequency square F1 of order n based on the set {s1,s2, ..., sm} with frequency vector (λ1, λ2, ...,λm) and a frequency square F2, also of order n, based on the set {t1,t2, ..., tk} with frequency vector (μ1, μ2, ...,μk) are orthogonal if every ordered pair (si, tj) appears precisely λiμj times when F1 and F2 are superimposed. Hall triple systems (HTSs) are Steiner triple systems (STSs) (but the blocks are called lines) with the property that the substructure generated by two intersecting lines is isomorphic to the finite affine plane AG(2,3). Any affine space AG(n,3) gives an example of an HTS. Such an HTS is an affine HTS. Nonaffine HTSs also exist. The number of points of an HTS is 3m for some integer m ≥ 2. Nonaffine HTSs exist for any m ≥ 4 and do not exist for m = 2 or 3. Every Steiner triple system is equivalent to a Steiner quasigroup (idempotent, commutative and satisfying (xy)y = x for all x and y). A Hall triple system is equivalent to a Steiner quasigroup which is distributive, that is, satisfies a(xy) = (ax)(ay) for all a,x,y in the quasigroup. Let S be a set of 2n elements. A Howell design, H(s,2n) (on symbol set S) is an s × s array such that: Each cell of the array is either empty or contains an unordered pair from S, Each symbol occurs exactly once in each row and column of the array, and Every unordered pair of symbols occurs in at most one cell of the array. An example of an H(4,6) is