<html><head><title>[design] 4 Determining basic properties of block designs</title></head>
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<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 element
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</strong> <var>r</var> if,
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 vector <var>L</var>
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>
<p>
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<P>
<address>design manual<br>November 2006
</address></body></html>
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