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<html><head><title>[design] 4 Determining basic properties of block designs</title></head>
<body text="#000000" bgcolor="#ffffff"> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP003.htm">Previous</a>] [<a href ="CHAP005.htm">Nex <h1>4 Determining basic properties of block designs</h1><p> <P> <H3>Sections</H3> <oL> <li> <A HREF="CHAP004.htm#SECT001">The functions for basic properties</a> </ol><p> <p> <p> <h2><a name="SECT001">4.1 The functions for basic properties</a></h2> <p><p> <a name = "SSEC001.1"></a> <li><code>IsBlockDesign( </code><var>obj</var><code> )</code> <p> This boolean function returns <code>true</code> if and only if <var>obj</var>, which can be an object of arbitrary type, is a block design. <p> <pre> gap> IsBlockDesign(5); false gap> IsBlockDesign( BlockDesign(2,[[1],[1,2],[1,2]]) ); true </pre> <p> <a name = "SSEC001.2"></a> <li><code>IsBinaryBlockDesign( </code><var>D</var><code> )</code> <p> This boolean function returns <code>true</code> if and only if the block design <var>D</var> is <strong>binary</strong>, that is, if no block of <var>D</var> has a repeated element. <p> <pre> gap> IsBinaryBlockDesign( BlockDesign(2,[[1],[1,2],[1,2]]) ); true gap> IsBinaryBlockDesign( BlockDesign(2,[[1],[1,2],[1,2,2]]) ); false </pre> <p> <a name = "SSEC001.3"></a> <li><code>IsSimpleBlockDesign( </code><var>D</var><code> )</code> <p> This boolean function returns <code>true</code> if and only if the block design <var>D</var> is <strong>simple</strong>, that is, if no block of <var>D</var> is repeated. <p> <pre> gap> IsSimpleBlockDesign( BlockDesign(2,[[1],[1,2],[1,2]]) ); false gap> IsSimpleBlockDesign( BlockDesign(2,[[1],[1,2],[1,2,2]]) ); true </pre> <p> <a name = "SSEC001.4"></a> <li><code>IsConnectedBlockDesign( </code><var>D</var><code> )</code> <p> This boolean function returns <code>true</code> if and only if the block design <var>D</var> is <strong>connected</strong>, that is, if its incidence graph is a connected graph. <p> <pre> gap> IsConnectedBlockDesign( BlockDesign(2,[[1],[2]]) ); false gap> IsConnectedBlockDesign( BlockDesign(2,[[1,2]]) ); true </pre> <p> <a name = "SSEC001.5"></a> <li><code>BlockDesignPoints( </code><var>D</var><code> )</code> <p> This function returns the set of points of the block design <var>D</var>, that is <code>[1..</code><var>D</var><code>.v]</code>. The returned result is immutable. <p> <pre> 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 ] </pre> <p> <a name = "SSEC001.6"></a> <li><code>NrBlockDesignPoints( </code><var>D</var><code> )</code> <p> This function returns the number of points of the block design <var>D</var>. <p> <pre> 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 </pre> <p> <a name = "SSEC001.7"></a> <li><code>BlockDesignBlocks( </code><var>D</var><code> )</code> <p> This function returns the (sorted) list of blocks of the block design <var>D</var>. The returned result is immutable. <p> <pre> 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 ] ] </pre> <p> <a name = "SSEC001.8"></a> <li><code>NrBlockDesignBlocks( </code><var>D</var><code> )</code> <p> This function returns the number of blocks of the block design <var>D</var>. <p> <pre> 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 </pre> <p> <a name = "SSEC001.9"></a> <li><code>BlockSizes( </code><var>D</var><code> )</code> <p> This function returns the set of sizes (actually list-lengths) of the blocks of the block design <var>D</var>. <p> <pre> gap> BlockSizes( BlockDesign(3,[[1],[1,2,2],[1,2,3],[2],[3]]) ); [ 1, 3 ] </pre> <p> <a name = "SSEC001.10"></a> <li><code>BlockNumbers( </code><var>D</var><code> )</code> <p> Let <var>D</var> be a block design. Then this function returns a list of the same length as <code>BlockSizes(</code><var>D</var><code>)</code>, such that the <var>i</var>-th elem of this returned list is the number of blocks of <var>D</var> of size <code>BlockSizes(</code><var>D</var><code>)[</code><var>i</var><code>]</code>. <p> <pre> 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 ] </pre> <p> <a name = "SSEC001.11"></a> <li><code>ReplicationNumber( </code><var>D</var><code> )</code> <p> If the block design <var>D</var> is equireplicate, then this function returns its replication number; otherwise <code>fail</code> is returned. <p> A block design <var>D</var> is <strong>equireplicate</strong> with <strong>replication number</stro for every point <var>x</var> of <var>D</var>, <var>r</var> is equal to the sum over the blocks of the multiplicity of <var>x</var> in a block. For a binary block design this is the same as saying that each point <var>x</var> is contained in exactly <var>r</var> blocks. <p> <pre> 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 </pre> <p> <a name = "SSEC001.12"></a> <li><code>PairwiseBalancedLambda( </code><var>D</var><code> )</code> <p> A binary block design <var>D</var> is <strong>pairwise balanced</strong> if <var>D</var> has at least two points and every pair of distinct points is contained in exactly <var>lambda</var> blocks, for some positive constant <var>lambda</var>. <p> Given a binary block design <var>D</var>, this function returns <code>fail</code> if <var>D</var> is not pairwise balanced, and otherwise the positive constant <var>lambda</var> such that every pair of distinct points of <var>D</var> is in exactly <var>lambda</var> blocks. <p> <pre> 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 </pre> <p> <a name = "SSEC001.13"></a> <li><code>TSubsetLambdasVector( </code><var>D</var><code>, </code><var>t</var><code> )</code> <p> Let <var>D</var> be a block design, <var>t</var> a non-negative integer, and <code></code><var>v</var><code>=</code><var>D</var><code>.v</code>. Then this function returns an integer v whose positions correspond to the <var>t</var>-subsets of <var>{1,...,v}</var>. The <var>i</var>-th element of <var>L</var> is the sum over all blocks <var>B</var> of <var>D</var> of the number of times the <var>i</var>-th <var>t</var>-subset (in lexicographic order) is contained in <var>B</var>. (For example, if <var>t=2</var> and <var>B=[1,1,2,3,3,4]</var>, then <var>B</var> contains <var>[1,2]</var> twice, <var>[1,3]</var> four times, <var>[1,4]</var> twice, <var>[2,3]</var> twice, <var>[2,4]</var> once, and <var>[3,4]</var> twice.) In particular, if <var>D</var> is binary then <var>L[i]</var> is simply the number of blocks of <var>D</var> containing the <var>i</var>-th <var>t</var>-subset (in lexicographic order). <p> <pre> 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 ] </pre> <p> <a name = "SSEC001.14"></a> <li><code>AllTDesignLambdas( </code><var>D</var><code> )</code> <p> If the block design <var>D</var> is not a <var>t</var>-design for some <var>tge0</var> then this function returns an empty list. Otherwise <var>D</var> is a binary block design with constant block size <var>k</var>, say, and this function returns a list <var>L</var> of length <var>T+1</var>, where <var>T</var> is the maximum <var>tlek</var> such that <var>D</var> is a <var>t</var>-design, and, for <var>i=1,...,T+1</var>, <var>L[i]</var> is equal to the (constant) number of blocks of <var>D</var> containing an <var>(i-1)</var>-subset of the point-set of <var>D</var>. The returned result is immutable. <p> <pre> gap> AllTDesignLambdas(PGPointFlatBlockDesign(3,2,1)); [ 35, 7, 1 ] </pre> <p> <a name = "SSEC001.15"></a> <li><code>AffineResolvableMu( </code><var>D</var><code> )</code> <p> A block design is <strong>affine resolvable</strong> if the design is resolvable and any two blocks not in the same parallel class of a resolution meet in a constant number <var>mu</var> of points. <p> If the block design <var>D</var> is affine resolvable, then this function returns its value of <var>mu</var>; otherwise <code>fail</code> is returned. <p> The value 0 is returned if, and only if, <var>D</var> consists of a single parallel class. <p> <pre> 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 </pre> <p> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP003.htm">Previous</a>] [<a href ="CHAP005.htm">Nex <P> <address>design manual<br>November 2024 </address></body></html> ¤ Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. 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