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% %A basic.tex DESIGN documentation Leonard Soicher % % % \def\DESIGN{\sf DESIGN} \def\GRAPE{\sf GRAPE} \def\nauty{\it nauty} \def\Aut{{\rm Aut}\,} \def\x{\times} \Chapter{Determining basic properties of block designs} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{The functions for basic properties} \>IsBlockDesign( <obj> ) This boolean function returns `true' if and only if an object of arbitrary type, is a block design. \beginexample gap> IsBlockDesign(5); false gap> IsBlockDesign( BlockDesign(2,[[1],[1,2],[1,2]]) ); true \endexample \>IsBinaryBlockDesign( <D> ) This boolean function returns `true' if and only if the block design <D> is *binary*, that is, if no block of <D> has a repeated element. \beginexample gap> IsBinaryBlockDesign( BlockDesign(2,[[1],[1,2],[1,2]]) ); true gap> IsBinaryBlockDesign( BlockDesign(2,[[1],[1,2],[1,2,2]]) ); false \endexample \>IsSimpleBlockDesign( <D> ) This boolean function returns `true' if and only if the block design <D> is *simple*, that is, if no block of <D> is repeated. \beginexample gap> IsSimpleBlockDesign( BlockDesign(2,[[1],[1,2],[1,2]]) ); false gap> IsSimpleBlockDesign( BlockDesign(2,[[1],[1,2],[1,2,2]]) ); true \endexample \>IsConnectedBlockDesign( <D> ) This boolean function returns `true' if and only if the block design <D> is *connected*, that is, if its incidence graph is a connected graph. \beginexample gap> IsConnectedBlockDesign( BlockDesign(2,[[1],[2]]) ); false gap> IsConnectedBlockDesign( BlockDesign(2,[[1,2]]) ); true \endexample \>BlockDesignPoints( <D> ) This function returns the set of points of the block design <D>, that is `[1..<D>.v]'. The returned result is immutable. \beginexample gap> D:=BlockDesign(3,[[1,2],[1,3],[2,3],[2,3]]); rec( isBlockDesign := true, v := 3, blocks := [ [ 1, 2 ], [ 1, 3 ], [ 2, 3 ], [ 2, 3 ] ] ) gap> BlockDesignPoints(D); [ 1 .. 3 ] \endexample \>NrBlockDesignPoints( <D> ) This function returns the number of points of the block design <D>. \beginexample gap> D:=BlockDesign(3,[[1,2],[1,3],[2,3],[2,3]]); rec( isBlockDesign := true, v := 3, blocks := [ [ 1, 2 ], [ 1, 3 ], [ 2, 3 ], [ 2, 3 ] ] ) gap> NrBlockDesignPoints(D); 3 \endexample \>BlockDesignBlocks( <D> ) This function returns the (sorted) list of blocks of the block design <D>. The returned result is immutable. \beginexample gap> D:=BlockDesign(3,[[1,2],[1,3],[2,3],[2,3]]); rec( isBlockDesign := true, v := 3, blocks := [ [ 1, 2 ], [ 1, 3 ], [ 2, 3 ], [ 2, 3 ] ] ) gap> BlockDesignBlocks(D); [ [ 1, 2 ], [ 1, 3 ], [ 2, 3 ], [ 2, 3 ] ] \endexample \>NrBlockDesignBlocks( <D> ) This function returns the number of blocks of the block design <D>. \beginexample gap> D:=BlockDesign(3,[[1,2],[1,3],[2,3],[2,3]]); rec( isBlockDesign := true, v := 3, blocks := [ [ 1, 2 ], [ 1, 3 ], [ 2, 3 ], [ 2, 3 ] ] ) gap> NrBlockDesignBlocks(D); 4 \endexample \>BlockSizes( <D> ) This function returns the set of sizes (actually list-lengths) of the blocks of the block design <D>. \beginexample gap> BlockSizes( BlockDesign(3,[[1],[1,2,2],[1,2,3],[2],[3]]) ); [ 1, 3 ] \endexample \>BlockNumbers( <D> ) Let <D> be a block design. Then this function returns a list of the same length as `BlockSizes(<D>)', such that the $i$-th element of this returned list is the number of blocks of <D> of size `BlockSizes(<D>)[$i$]'. \beginexample gap> D:=BlockDesign(3,[[1],[1,2,2],[1,2,3],[2],[3]]); rec( isBlockDesign := true, v := 3, blocks := [ [ 1 ], [ 1, 2, 2 ], [ 1, 2, 3 ], [ 2 ], [ 3 ] ] ) gap> BlockSizes(D); [ 1, 3 ] gap> BlockNumbers(D); [ 3, 2 ] \endexample \>ReplicationNumber( <D> ) If the block design <D> is equireplicate, then this function returns its replication number; otherwise `fail' is returned. A block design $D$ is *equireplicate* with *replication number* $r$ if, for every point $x$ of $D$, $r$ is equal to the sum over the blocks of the multiplicity of $x$ in a block. For a binary block design this is the same as saying that each point $x$ is contained in exactly $r$ blocks. \beginexample gap> ReplicationNumber(BlockDesign(4,[[1],[1,2],[2,3,3],[4,4]])); 2 gap> ReplicationNumber(BlockDesign(4,[[1],[1,2],[2,3],[4,4]])); fail \endexample \>PairwiseBalancedLambda( <D> ) A binary block design $D$ is *pairwise balanced* if $D$ has at least two points and every pair of distinct points is contained in exactly $\lambda$ blocks, for some positive constant $\lambda$. Given a binary block design <D>, this function returns `fail' if not pairwise balanced, and otherwise the positive constant $\lambda$ such that every pair of distinct points of <D> is in exactly $\lambda$ blocks. \beginexample gap> D:=BlockDesigns(rec(v:=10, blockSizes:=[3,4], > tSubsetStructure:=rec(t:=2,lambdas:=[1])))[1]; rec( isBlockDesign := true, v := 10, blocks := [ [ 1, 2, 3, 4 ], [ 1, 5, 6, 7 ], [ 1, 8, 9, 10 ], [ 2, 5, 10 ], [ 2, 6, 8 ], [ 2, 7, 9 ], [ 3, 5, 9 ], [ 3, 6, 10 ], [ 3, 7, 8 ], [ 4, 5, 8 ], [ 4, 6, 9 ], [ 4, 7, 10 ] ], tSubsetStructure := rec( t := 2, lambdas := [ 1 ] ), isBinary := true, isSimple := true, blockSizes := [ 3, 4 ], blockNumbers := [ 9, 3 ], autGroup := Group([ (5,6,7)(8,9,10), (2,3)(5,7)(8,10), (2,3,4)(5,7,6)(8,9,10), (2,3,4)(5,9,6,8,7,10), (2,6,9,3,7,10)(4,5,8) ]) ) gap> PairwiseBalancedLambda(D); 1 \endexample \>TSubsetLambdasVector( <D>, <t> ) Let <D> be a block design, <t> a non-negative integer, and `$v$=<D>.v'. Then this function returns an integer vector $L$ whose positions correspond to the <t>-subsets of $\{1,\ldots,v\}$. The $i$-th element of $L$ is the sum over all blocks $B$ of <D> of the number of times the $i$-th <t>-subset (in lexicographic order) is contained in $B$. (For example, if $t=2$ and $B=[1,1,2,3,3,4]$, then $B$ contains $[1,2]$ twice, $[1,3]$ four times, $[1,4]$ twice, $[2,3]$ twice, $[2,4]$ once, and $[3,4]$ twice.) In particular, if <D> is binary then $L[i]$ is simply the number of blocks of <D> containing the $i$-th <t>-subset (in lexicographic order). \beginexample gap> D:=BlockDesign(3,[[1],[1,2,2],[1,2,3],[2],[3]]);; gap> TSubsetLambdasVector(D,0); [ 5 ] gap> TSubsetLambdasVector(D,1); [ 3, 4, 2 ] gap> TSubsetLambdasVector(D,2); [ 3, 1, 1 ] gap> TSubsetLambdasVector(D,3); [ 1 ] \endexample \>AllTDesignLambdas( <D> ) If the block design <D> is not a $t$-design for some $t\ge 0$ then this function returns an empty list. Otherwise <D> is a binary block design with constant block size $k$, say, and this function returns a list $L$ of length $T+1$, where $T$ is the maximum $t\le k$ such that <D> is a $t$-design, and, for $i=1,\ldots,T+1$, $L[i]$ is equal to the (constant) number of blocks of <D> containing an $(i-1)$-subset of the point-set of <D>. The returned result is immutable. \beginexample gap> AllTDesignLambdas(PGPointFlatBlockDesign(3,2,1)); [ 35, 7, 1 ] \endexample \>AffineResolvableMu( <D> ) A block design is *affine resolvable* if the design is resolvable and any two blocks not in the same parallel class of a resolution meet in a constant number $\mu$ of points. If the block design <D> is affine resolvable, then this function returns its value of $\mu$; otherwise `fail' is returned. The value 0 is returned if, and only if, <D> consists of a single parallel class. \beginexample gap> P:=PGPointFlatBlockDesign(2,3,1);; # projective plane of order 3 gap> AffineResolvableMu(P); fail gap> A:=ResidualBlockDesign(P,P.blocks[1]);; # affine plane of order 3 gap> AffineResolvableMu(A); 1 \endexample ¤ Dauer der Verarbeitung: 0.2 Sekunden (vorverarbeitet) ¤*© Formatika GbR, Deutschland |
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2026-03-28