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<title>GAP (ALCO) - Chapter 4: Spherical and Projective Designs</title>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X82BCF4BA84BDEE9E">4.2-1 <span class="Heading">Jordan Design Categories</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X85F7CF2D7FDF58D7">4.6-1 JordanDesignBoseMesnerAlgebra</a></span>
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<span class="ContSS<span =">nbsp;&bsp;/>a =chtml#84"464JordanDesignKreinNumbers</><span>
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<span<>. span class"eading"Jacobi Polynomials/span>/>
<pOne  of  spherical  designs assets   idempotents  thespherical  projectivecasescan  studiedtogether Jacobi , withsuitableparameterschosenforjava.lang.StringIndexOutOfBoundsException: Range [256, 255) out of bounds for length 292
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<h3>4 <span class="Heading">Spherical and Projective Designs</span></h3>

or  isjava.lang.StringIndexOutOfBoundsException: Range [48, 47) out of bounds for length 510

<p>Specifically, the primitive idempotents of simple Euclidean Jordan algebras of rank <span class="Math">2</span> have the geometry of a sphere.

<p>The tools below allow one to construct a <strong class="pkg">GAP</strong> object to represent a design and collect various computed attributes. Constructing a design and its parameters using these tools does<pan class">><span "APinputb=Indeterminatejava.lang.StringIndexOutOfBoundsException: Range [91, 90) out of bounds for length 106

<p><a id="X79A71E957D5B9755" name="X79A71E957D5B9755"></a></p>

<h4>4.1 <span[1/*a-/*,12*+1*+1]

<pOne  of studying   projective together    primitive isboth  spherical and casescanbe studied  usingJacobipolynomials  suitableparameterschosen  the simpleEuclidean Jordanalgebra<p

<p><a id="X872DC5F085155040" name="X872DC5F085155040"></a></p>

<h5>4.1-1 JacobiPolynomial

<div class=
<returns polynomial span class=="ath">P_k^{(,)}(<span of degree < <var class="Arg"></var and type <var class="Arg">(a,b)</var> as defined in <a href="chapBib.html#biBabramowitz_handbook_1972">[AS72, chap. 22]</a>. The argument <var class="Arg">k</var> must be a non-negative integer

<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">a := Indeterminate(Rationals, "a");; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">b := Indeterminate(Rationals, "b");; </span>
<span class="GAPprompt">gap<div class="func"<tableclass""width="100%><><td class=tdleft"< class"func"&#227 R_k_epsiloncode(varclass"Arg"k>,<class=Argepsilon/var> <class"">rank/>  ""degreevar,var"rg></var>)/><td =">&bspfunction;<td<tr/></>
<span=">&;/> <span class=G"JacobiPolynomial,,b;/pan
[ 1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">JacobiPolynomial(1,a,b);</span>
[ 1/2*a-1/2*b, 1/2*a+1/2*b+1 ]
<span
1/2**+/**x1/*-/*+<pre<div

<p><<p><a"" name"">/>/

<5>.2< =Heading"RenormalizedJacobiPolynomials<span></5java.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Q_k_epsilon</code>( <var class="Arg">k</var>, <var class="Arg">epsilon</var>, <var class="Arg">rank</var>, <var class="Arg">degree</var>, <var class="Arg">x</var> )</td><td class=<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSphericalJordanDesign</code>/td><tdclass"dright"(&bspfilternbsp;<td<tr<table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; R_k_epsilon</code>( <This  aJordan in of  java.lang.StringIndexOutOfBoundsException: Range [95, 91) out of bounds for length 173
<p>These functions return <design with rank 3 and degree 8  3and &;

<p><a id=tjava.lang.StringIndexOutOfBoundsException: Range [4, 5) out of bounds for length 4

<h4>4.2 <span class="Heading">Jordan Designs</spanfalse

<p>The <strong class="pkg">ALCO</strong> package defines new categories within <code class="code">IsObject</code> in order to constructjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4

<p><a id="java.lang.StringIndexOutOfBoundsException: Range [0, 27) out of bounds for length 4

<h5>4.2-1 <span

<div class="func"><table class="func" width="100%"><tr><td class=
< classfunc>table class=""width=100"<tr>td class=tdleft"<code class=func"‣; IsSphericalJordanDesign</ode></d< =">&f&;/>/><table>>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><java.lang.StringIndexOutOfBoundsException: Range [0, 79) out of bounds for length 0
pThese filters determine whetheranobjectis  Jordandesignandwhether design is constructed in a spherical or projective manifold of Jordan primitive idempotents.</p>

<p><a id="X7FD006D77B3C98E2" name="X7FD006D77B3C98E2"></a></p>

<h5>4.2-2 JordanDesignByParameters</h5>

<div class=<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ JordanDesignDegree</code>( <>The  and  degree an object satisfying filter<code class""IsJordanDesign<>arestored attributes</>
<p>This function constructs a Jordan design in the manifold of Jordan primitive idempotents of rank <var

< class="xample">pre<class=GAPprompt>>/> <panclass=GAPinput> : JordanDesignByParameters,8;/span
<design with rank 3 and degree
<"GAPpromptgapgt<span <panclass="APinput">IsJordanDesign()</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSphericalJordanDesign(D);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsProjectiveJordanDesign(D);</span>
java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 4
<;</span> <spanclass="APinput">JordanDesignByParameters(4,8);</span>
fail
<span class=
fail</< class"example">pre< =&;< =Gjava.lang.StringIndexOutOfBoundsException: Range [87, 86) out of bounds for length 130

<< id="=java.lang.StringIndexOutOfBoundsException: Range [53, 52) out of bounds for length 62

<h5>4.2-3 <span class="Heading">Jordan Rank and Degree</span></h5=Gjava.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 97

divclass"">table=f ="00%">tr<td class=tdleft>code class="func&8227; JordanDesignRank</ode>(< class="Arg><var </>td class="(nbsp;attribute <td</></><div>
<">‣ JordanDesignDegree<( var ="></ar)/>tdclass"">&bspa )/d>/r<table<>
<p>The rank and degree of an object satisfying filter <code class="code">java.lang.StringIndexOutOfBoundsException: Index 84 out of bounds for length 29

<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">D := JordanDesignByParameters(3,8);</span>
& and degree 8&t
<span class="GAPprompt">gap><
[3 8 ]/re>/>

<p>" name="X87D80CFF7F4FD7D6"<a<p>

<h5>4.2-4 JordanDesignQPolynomials</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; JordanDesignQPolynomials</java.lang.StringIndexOutOfBoundsException: Range [0, 131) out of bounds for length 39
<p>Many java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21

<div class="xample"><pre>span =G">/ spanclass"APinput> := JordanDesignByParameters(38;/>
&t withrank 3anddegree8g;
<span class="[    /39,    5/273,   1/19,       0],
<span class="GAPprompt">gap></span>   [         5/91,     1/91,    1/728  1/2376 ]]
<span class="GAPprompt">gap></span> <span class=span class="GAPprompt">gap></span> <span class="GAPinput">for j in [1..4] do Display(Sum(List([1..4<span class="APprompt">></span> <span class="GAPinput">f[j][i]*JordanDesignQPolynomials(D ,0, 0, 0 ]
function( k java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="GAPprompt
[90 -585, 819 ]
<span class="
[ 90,-585, 819 pre></div>

<p><a id="X80A1BCF87B87D691" name="X80A1BCF87B87D691"></a></p>

<4.2-5 JordanDesignConnectionCoefficients/>

<class="unc"<tablejava.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 242
<>connectionofdesignvar"rg>D</var>determine which linear combinations of <code class"code>(varclass"D<var)/code>java.lang.StringIndexOutOfBoundsException: Range [190, 189) out of bounds for length 545

<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">D := JordanDesignByParameters(3,8);</span>
<design with rank 3 and degree 8>
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignConnectionCoefficients(D);</span>
function( s ) ... end
<span class="GAPprompt">gap></span> <span class="GAPinput">f :=
[ [1,               ,        0]
  [      1/3,     1/39,        0,        0
  [     5/39,    5/273,    1/819,        0 ],
  [     5/91,iv class="unc"<class"uncwidth=100%"><classtdleft><class=>#;JordanDesignAngleSet( var=""></> <td<d class=tdright>n;attribute;<td>/r<table/>
<pFor a  <var class=>D/>  an  ,records theangle var ="">/ asan  <odeclass="code"</code> The anglesetvarclass""><var mustbealistofrealvalued in =code></>intheinterval<span =M>0 1<. Note that  varclass=Arg></>  irrationalfor ltdoes   anordering  intheinterval  is  . note that angle set cannot      attribute ofthedesign<p
<java.lang.StringIndexOutOfBoundsException: Range [22, 5) out of bounds for length 114
[1 ,0 0]
[ 0, 1, 0, 0 ]
[ 0, 0, 1, 0 ]
[,0,0,1 </pre</div

<p><a id="X87DA51657F7EC947" name="X87DA51657F7EC947"></a></p>

<h4>4.3 <span class="Heading">Designslt;designwith rank4and degree 4&t

<p>The angle set of a design is the set of alllt;esignwithrank 4,degree4,andangle [19, 1/ >

<p9 3 <<div

<h5>4.3-1 IsJordanDesignWithAngleSetjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsJordanDesignWithAngleSet</code></td><td class="tdright">( filter )</td></tr></table></div>
p filteridentifies   that <code class""IsJordanDesign<code   withan set</p>

<p><a id="X78AC6815875A2024" name="X78AC6815875A2024"></a></p>

<h5>4.3-2 <span class="Heading">Design Angle Sets</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; JordanDesignAddAngleSet</code>( <var class="Arg">D</var>, <var class
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; JordanDesignAngleSet</code>( <var class="Arg">D</var> )</tdlt;esign ,4   set19 / &;
<p>For a design <var class="Arg">D</var> without an angle set, records 1/,1/ </re></div>

<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">D := java.lang.StringIndexOutOfBoundsException: Index 99 out of bounds for length 0
&with rank   4gt
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignAddAngleSet(D, [2]);</span>
fail
<span class"GAPprompt">gap>/span><span ="GAPinput"D;/span>
&java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 0
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignAddAngleSet(D, [1/3,1/9]);</span>
&;esign  rank 4,degree4  angleset[ /9, 13 >java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
<pan class="gap&;</pan>span class=GAPinput>JordanDesignAngleSet()</pan>
[ 1/9, 1/3 ]</pre></div>

<p><a id="X840BF5428457449B" name="X840BF5428457449B"></a></p>

<h5>4.3-3 JordanDesignByAngleSet</h5java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1

<div class="func"><table class="func" width0
< degree given by<class=Arg>ank<var and <var class=">egree<var, with angle set < ="rg></>  same onanglejava.lang.StringIndexOutOfBoundsException: Range [215, 214) out of bounds for length 389

<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">D := JordanDesignByAngleSet(4, 4, [1/3, 1/9]);</span>
<design with rank 4, degree 4, and angle set [ 1/9, 1/3 ]>
<span
[ 1/,1/ ]/re><>

<p><a id="X7923181A81406714h5.-5 JordanDesignNormalizedIndicatorCoefficients<h5>

<>43- JordanDesignNormalizedAnnihilatorPolynomialh5

<divclass="unc"> classf" width="00"<>tdclass=tdleft"<code class"">#8227; JordanDesignNormalizedAnnihilatorPolynomial>(< class""></>)/>< ="tdright>&;attribute&;)/d</tr>/></iv>
<Thenormalized polynomialisjava.lang.StringIndexOutOfBoundsException: Range [59, 58) out of bounds for length 501

<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">D := JordanDesignByAngleSet(4, 4, [1/3, 1/9]);</span>
&;  4,degree4,andangle set[
<span class="GAPprompt">gap></span> <span class class=GAPpromptgapg<span span =>(List[.3,i&; [i]JordanDesignQPolynomials)i1))<span>
[116 -4,27/16 java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
<span class="GAPprompt">gap></span> <span class="
0
<span class="GAPprompt">gap></span> <span classjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
0
<span class="GAPprompt">gap></span> <p>This filter determines whether the normalized indicator coefficients of a design are
1</pre></div>

java.lang.StringIndexOutOfBoundsException: Index 168 out of bounds for length 62

<h5>4.3-5 JordanDesignNormalizedIndicatorCoefficients</h5>

<div class
<p>The normalized indicator=e>pre<spanclass">&t;/><spanclass="APinput> =JordanDesignByAngleSet4 131/];/>

<div class="example"><pre><span span class="GAPprompt">gap></span"">JordanDesignSpecialBound()<span>
<64<>/>
<span class="java.lang.StringIndexOutOfBoundsException: Range [0, 22) out of bounds for length 0
[ 1/64, 7/960, 9/3520 ]
< =GAPprompt>&t<span><class"APinput">Sum(ist[.3]  -gt []()(-))<span>
[ 1/16, -3/4, 27/16 ]
<span class="p>Many more propertiesof a design with an angle set can be computed once the cardinality of the design is also known. In what follows let <span class="Math">v</span> be the cardinality of a design and let <span class="Math">s</span> be the cardinality of the angle set <span class="Math">Ajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[ 1/16, -3/4, 27/16 ]</

<p>iv=func<table class="func" width"100%>tr><td ="tdleft"code ="unc>#8227; JordanDesignAddCardinality</code>( <var class="Arg">D</var>, <var class="Arg">v</var> )</td><td class="tdright">( operation )</td></tr></table></div>

<h5>4.3-6 IsJordanDesignWithPositiveIndicatorCoefficients</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsJordanDesignWithPositiveIndicatorCoefficients</code></td><td class=>  finite seteachdesignhascardinalityWhen cardinality   for object var=Arg><>thatsatisfiescode"ode">sJordanDesign(var class""><><> cardinality is stored   java.lang.StringIndexOutOfBoundsException: Range [247, 246) out of bounds for length 755
<p>This filter determines whether the normalized indicator coefficients of a design are positive, which has significance for certain theorems about designs.</p>

<p><a id="X858B619F7E32D5EB" name="X858B619F7E32D5EB"></a></p>

<h5>4.3-7 JordanDesignSpecialBound</h5>

<div class="func"><table class="func" width="100%"><tr><td
<> a java.lang.StringIndexOutOfBoundsException: Range [45, 43) out of bounds for length 335

<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">D := JordanDesignByAngleSet(4, 4, [1/3,1/9]);</span>
<design with rank 4, degree 4, and angle set [span class""gapgt;/> <class"APinput">HasJordanDesignCardinality();<>
< =GAPprompt>gapg;/> spanclass=>IsJordanDesignWithPositiveIndicatorCoefficientsD;<span>
true
<span class="GAPprompt">gap></span> <span span class="GAPprompt">gap></span> <span class()<span>
<pre</>

<p><a id="X803F6A9986E27413"p><aid=X7CA9D3F585A7CBC3nameXC">/><pjava.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62

<

<p>p>  in< "></code> <a href="hap4html">span =">.-/>)    of design    set the cardinality   canwhether design   bound       angleand   the  <p>

<p><a id="X7F730A1C7FC9D987" name="X7F730A1C7FC9D987"></a></p>

<h5>4.4-1 <span class="Heading">

<div class=
div class"">table class="unc" idth"00">tr><td""<=func#;</  ""<><class"Arg">v</ )<td< ">&n;<>/<>/>
<"">="func java.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 231
<p>As a finite set, each design has a cardinality. When this cardinality is known for an object <var class="Arg">D</var> that satisfies <code class="code">IsJordanDesign(<var class="Arg">D</var>)</code>, the cardinality is stored as ( ="D/var>)/ode>.<p>

<class"xample">preclass=">gap&t<span> span class=GAPinput"D: (4 [/3,9)<span
<design with rank 4, degree 4, and angle set [ 1/9, 1/3 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">HasJordanDesignCardinality(D);</span>
false
<span class="GAPprompt">gap></span[ , -48, 108  ]
&t2design  withrank 4  4  64,    /,1/ &;
<span class="java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 14
<pre<div>

<p><a id>4.4-4JordanDesignIndicatorCoefficients</h5

<h5>4.-2 <span class=Heading"Designs at theSpecial Bound</span></h5>

<div class="func"><table class<> indicatorcoefficientsfor design var class"rg"D/var>are defined bymultiplying<code class="ode">JordanDesignNormalizedIndicatorCoefficients(var=""</ar)> =c"JordanDesignCardinality(<var class="Arg">D</var>)</code>. These indicator coefficients are often useful for directly determining the strength
<p>As described in <code class="func">span class="GAPprompt">gap></span> <span class="GAPinput&<;/  Gjava.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 106

< id"80java.lang.StringIndexOutOfBoundsException: Range [29, 27) out of bounds for length 62

<>4- />

<div class="func"><table class="func" width="100%
<">JordanDesignNormalizedAnnihilatorPolynomial(<var="></>/> <class">< =></)/./>

<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">D := JordanDesignByAngleSet(4, 4, [1/3, 1/9]);; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignAddCardinality(D, 64);; D;</span>

<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignAnnihilatorPolynomial(D);</span>
[ 4, -48, 108 ]
<
64pre/

<>a=X =>a/>

<>4- /

<div class="func"><table< Gjava.lang.StringIndexOutOfBoundsException: Range [24, 22) out of bounds for length 93
<p>The indicator

<div class="java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 0
<2-design with rank 4, degree 4, cardinality 64
<java.lang.StringIndexOutOfBoundsException: Range [22, 5) out of bounds for length 106
[ 1, 7/15, 9/55 ]</pre></div>

<p><a id="X7C4ADEEA8355A774" name="X7C4ADEEA8355A774"< ="unc"<java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 196

<h5>4.4-5 <span class="Heading">Design Strength</span></h5>

<class"< = =100%<> java.lang.StringIndexOutOfBoundsException: Range [65, 64) out of bounds for length 203
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; JordanDesignStrength</code>( <var class="Arg">Djava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>The <span class="Math">t</span>-design is a design with the following special property: the integral of any degree <span class="Math">t</span> polynomial over the sphere or projective space containing the design is equal to the average

<p>For a design <var class="Arg">D</var> that satisfies <code class="code">IsJordanDesignWithPositiveIndicatorCoefficients</code>, <code class="code">IsJordanDesignWithCardinality</code>, and <code class="code">IsSpecialBoundJordanDesign</code>, we can compute the strength <span class="Math">t</span> of the design using a theorem given in <a href="

<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">D := JordanDesignByAngleSet(4,4, [1/3,1/9]);</span>
&tdesignrank , ,angle java.lang.StringIndexOutOfBoundsException: Range [48, 47) out of bounds for length 64
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignAddCardinality(D, 64);</span>
<2-design with rank 4, degree 4, cardinality 64, and angle set [ 1/9, 1/3 ]>
<span class="GAPprompt">gap></span> <
true
class="gt;/ span ="D;</>
2</pre></div>

<p><a id="span =G"gapg;<>< =Gjava.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 95

<h5

<div class="<h4>4.6< =Heading>    </><h4>
<div class="func"><table class="p> a   <panclass="t \ge   <  itadmits java.lang.StringIndexOutOfBoundsException: Range [108, 107) out of bounds for length 440
< =f>table class"unc"width=%<>tdclass"">code =func&8227;IsTightJordanDesign</ode>/>td ="dright"(nbspfilter&;)/d<tr>/able/>
<p>These filters identify various special categories of designs that satisfy <code class="func">IsJordanDesignWithStrength</code> (<a href="chap4.html#X7C4ADEEA8355A774"><span class="RefLink">4.4-5</span></a>). In what follows recall that <span class="Math">t</span> denotes the strength of the design and <span class="Math">s</span> denotes the cardinality of the angle set <span class="Math">A</java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

<p>A design admits a <em>regular scheme</em> when <span class="Math">t \ge s - 1 </span>. The filter <code class="code">IsRegularSchemeJordanDesign</code> returns true when both <span class="Math">t</span> and <span class="Math">s</span> are known and satisfy the regular scheme inequality given above.</p>

<p>A design admits an <em>association scheme</em> when <span class="Math">t \ge 2s - 2 </span>. The filter <code class="code">IsAssociationSchemeJordanDesign</code> returns true when both <span class="Math">t</span> and <span class="Math">s</span> are known and satisfy the association scheme inequality given above.</p>

<p>Finally, a design is <em>tight</em> when it satisfies <span class="Math">t = 2s - 1</span> for <span class="Math">0</span> in <span class="Math">A</span> or <span class="Math">t = 2

< A1,  java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14

<h4>4.5 <span class="Heading">Designsjava.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2

<p>a="83501A01792752CC" name"83501A01792752CC>/a>/>

<<h5>46- </h5>

<iv class""><able =func">‣ JordanDesignSubdegrees/code>(< class="Arg></ar <td<d class""(nbsp;ttributenbsp;)/td>/>/table>/>
<p>For a design <var class="Arg">D</var> withp>For adesignthatsatisfiescode =c></code,we also define  idempotent of the corresponding-Mesner a href".html#iBbannai_algebraic_2021>BBIT21 . 5357<a><p>

<div class="example"><pre><span class="GAPprompt">gap></lt2 with  4  4  64    /1/3]java.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 82
&;2-esign  4,degree4  64   set[ 19,13 &;
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignSubdegrees(D);</span>
[ 27, 36 ]</pre></div>

<p><a id="X7EE9F8D97A51FBF9" name="X7EE9F8D97A51FBF9(116*A1+(-/16)A2(9/16)A3

<h446 <span class"Heading">Designs Admitting an Association Scheme</span></h4>

<p>When a design satisfies <span <span class="GAPprompt">gap></span class"GAPinput">orAll((D), IsIdempotent);/>

<p><a id="85F7CF2D7FDF58D7" name="X85F7CF2D7FDF58D7"</a</>

<h5>4.6-1 JordanDesignBoseMesnerAlgebra</h5>

<iv class=func>table class=func"width=100%>tr>td ="dleft>code "">&#>8227; JordanDesignBoseMesnerAlgebracode class=Arg><var )/<dclass"tdright">&bsp;ttribute <td<tr/>/>
<p>For a design thatdiv =func">table =""width=100"<>< ="tdleft>code ="unc>#8227;JordanDesignIntersectionNumbers(< =A"D/> )</d><td =tdright"(nbsp;attribute )/><tr</table><div

<div class="java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 0
&t2design  rank 4 degree 4, cardinality 64, and angle set [ 1/9, 1/3 ]>
<span class="GAPprompt">gap></spanlt;2-design with rank4,degree 4  64,and  set [ 1/9,13 >
<algebraofdimension Rationalsjava.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
<span class="GAPprompt"<span class="GAPprompt">gap></span> <span class="GAPinput">p := JordanDesignIntersectionNumbers(D);;</span>
[A1 A2,  ]
<span class="GAPprompt">gap></span> <span class="GAPinput">One(Btrue</pre>/div>

true</java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p><a id="X7F95AE4A7B7B5B2B" java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

<h5>4.6-2 JordanDesignBoseMesnerIdempotentBasis</h5>

<iv ""<class="unc width="00%"><tr>td ="tdleft>code class="unc">8227 </code> <var ="Arg"D/ar </d< class"dright"(nbspattribute)/<tr<table<div
<p>For a design that satisfies <code classdiv ="xample"<>< "GAPprompt">&<span>< class"APinput"> =JordanDesignByAngleSet4,4,[3, 19);JordanDesignAddCardinality 64) ;</span>

<div class="example"><pre><span class="GAPprompt">gap></spanspan class=GAPprompt>;span <pan ="APinput"q=JordanDesignKreinNumbers(D);/
&;2withrank ,degree4 cardinality 64,and angle set[1/, 1/3]gt;
<span class="GAPprompt">gap></[  10, 16  , [16 20 0 ]  1 ,0  ]
<span class="GAPprompt">></span> <span class="GAPinput">od;</span>
(-5/64)*A1+(3/64)*A2+(27/64)*A3
(1</pre></div>
(1/64)*A1+(1/64)*A2+(1/64A"name=X87C0A2067F13F01A"<a</
<span class="APprompt"gap&t</span>span class=GAPinput>ForAll(ordanDesignBoseMesnerIdempotentBasis,IsIdempotent;/pan>
true</pre

<p><a id="X851F03427CFDC513" name="X851F03427CFDC513"></a></p>

<h5>4.6-3 JordanDesignIntersectionNumbers</h5>

<div class=func"><table class=func"width="100%"><tr><td class="tdleft"><code class="func">&# style='color: green'>8227; JordanDesignIntersectionNumbers</code>( <var class="Arg">D</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The intersection numbers <span class="Math">p^k_{i,j}</span class"APprompt"><span> < class=">a := Basis(JordanDesignBoseMesnerAlgebra());;/>

<div class="examplespan class=GAPprompt"gap><span < =GAPinput>[.3,i-gt [i =Sum1.] -<>
<2-esignwith  4,degree4 cardinality 64 and angle set[19 / ]gt;
<span class="GAPprompt">gap></span> <span class="GAPinput">A := BasisVectors(Basis(java.lang.StringIndexOutOfBoundsException: Index 104 out of bounds for length 16
<span class="GAPprompt">gap></span> <span class="GAPinput">p := JordanDesignIntersectionNumbers(D);;</span>
<span class="GAPprompt">gap></span> <span class="
true</>/div>

<p><a id="X84E0CAFE83145B3F" name="X84E0CAFE83145B3F"></a></p>

<h5>4.6-4 JordanDesignKreinNumbers</h5>

< =f>table class"" width"100%>tr><td =t"< class=">#227 </code> var class=Arg"D/ </<class"dright>&;attribute&bsp;<td</r><table>/div>
<p>The <2-design rank4 degree 4,cardinality 64  angleset[1/,1/3 ]gt

<div class="example"><pre><spanspan class="APprompt">gap></span> <span class="GAPinput">e := JordanDesignBoseMesnerIdempotentBasis(D);;</span>
<-   4,  4,cardinality 64,and  set [ 1/9,1/ >
<span class="GAPprompt">gap></span> <span class="GAPinput">q := JordanDesignKreinNumbers(D);; < =GAPprompt>gt;/ < "GAPinput"Sum[.3,j-gt;JordanDesignSecondEigenmatrix(D)[i][j]*a[j]));</span>
<span class="GAPprompt">gap></spanspan class"">gap&t<span <panclass="APinput"JordanDesignFirstEigenmatrix) JordanDesignSecondEigenmatrix)<span
[[[10 16,1 ],[16,20, 0 ,  1,0,0 ],
  [ [ 12, 15, 0 ], [ 15, 20, 1 ], [ 0, 1, 0true/><div>
    27 0,0 ]  0,36 0]  0,,1]] java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
  </pre<5>.- JordanDesignMultiplicities<h5java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41

<p<a id"87C0A2067F13F01AC0A2067F13F01A" name="X87C0A2067F13F01A"></a></p>

<h5>4.6-5 JordanDesignFirstEigenmatrix</h5>

<div class="func"><java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 0
<p> describe in <a href="chapBib.html#biBbannai_algebraic_2021">[BBIT21, p. 58]</a>, the first eigenmatrix of a Bose-Mesner algebra <span java.lang.StringIndexOutOfBoundsException: Index 145 out of bounds for length 82

<div class="example"><pre><span class="GAPprompt">gap& ,36,1<></>
&2-java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 82
<span class=
<=java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 116
<span class="GAPprompt">gap></span> <span class="GAPinput">ForAll([1.java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="GAPprompt">></span> <span class="GAPinput">JordanDesignFirstEigenmatrix(D)[i][j]*e[j]));</span>
true</pre></div>

<p><a id="X7FC9FDCC7DD82286" name="X7FC9FDCC7DD82286"></a></p>

<h5>4.6-6 <2-design with rank 4 4  64  angle set [ 1/9, 1/3 ]>

<div class=func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ JordanDesignSecondEigenmatrix<code>( <var class="Arg">D</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>As describe in <a href="

<div class="example"><pre><span class="h5>4.6-9 JordanDesignReducedAdjacencyMatrices>
<2-design with rank 4, degree 4, <divclass="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ JordanDesignReducedAdjacencyMatrices</code>( <var class="Arg">D</var>p>As definedin< href="chapBib.html#biBcameron_designs_1991">[CVL91, p. 201]</a>, the reduced adjacency matrices multiply with the same java.lang.StringIndexOutOfBoundsException: Index 145 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 230 out of bounds for length 115
<span class="GAPprompt">gap></span> <span class="GAPinput">e := JordanDesignBoseMesnerIdempotentBasis(D);;</span>
<span class="GAPprompt">gap>div class="xample"<><span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignByAngleSet(2, 1, [0,1/2]);;</span>
<span class="GAPprompt">></span> <span class="GAPinput">Sum([1..3], j -> JordanDesignSecondEigenmatrix(D)[i][j]*a[j]));</span>

<panclass="APprompt>gap>/span> <span class="APinput>JordanDesignFirstEigenmatrixD  InverseJordanDesignSecondEigenmatrixD)/span>
<span">gt<java.lang.StringIndexOutOfBoundsException: Range [35, 34) out of bounds for length 94
true</pre></div>

<p><a id="X7FB944C182A2FBE3" nameltT 3design rank2 degree  ,and set

<h5>4.6-7 span class="GAPprompt">gap/pan> span =""(,4 [,/];;</span>

<div<td class"dleft> class"func>&#227 JordanDesignMultiplicities/> < class="rg><var>)/>td class"dright>nbspattribute&;)></r<table>/iv
<p>As describe in <a href="chapBib.html#biBbannai_algebraic_2021">[BBIT21, pp. 58-59]</a>, the design multiplicy <span class="Math">m_i</span><Tight3-design with  2, degree 4, cardinality 10, and angle set

< =example>pre< =GAPprompt>gap&t/span span =GAPinput>D :=JordanDesignByAngleSet,,13,1/9]);; JordanDesignAddCardinality,64); D;/span>
&;2designwith 4,degree4 cardinality 64 and angle  [19 / >java.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 82
<span class="GAPprompt"><span class="GAPprompt">gap/> span class"APinput">(3, 2 [/];;<span>
[27 36  ]</pre<div

<p><a id="span ="APprompt>&t/>< =G"JordanDesignByAngleSet4,2 [,/];;/>

<h5>4.6-8 DesignValencies</h5>

<div class="func"><table class="func" width="100%"><tr><td class-with ,,java.lang.StringIndexOutOfBoundsException: Range [54, 53) out of bounds for length 71
<p>As describe in <a href="chapBib.html#biBbannai_algebraic_2021">[BBIT21, pp. 55, 59< =G"gap;</pan>span =">(last(last);/>

<div classjava.lang.StringIndexOutOfBoundsException: Range [24, 22) out of bounds for length 106
<2-design with rank 4, degree 4, cardinality 64, and angle set [ 1/9, 1/3 ]<-java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 83
<class""gapgt;<>span=GAPinputDesignValenciesD</pan
[ 27, 36, 1 ]</pre></div>

<p><a id="X788831797E45CE02" name=lt;ight 3-  rank,degree ,cardinality165, and

<-9 JordanDesignReducedAdjacencyMatricesh5>

<java.lang.StringIndexOutOfBoundsException: Range [91, 4) out of bounds for length 244
pAsin<href.html#iBcameron_designs_1991 .]/,theadjacency  withsameconstants theadjacencymatricesfor  constructionjava.lang.StringIndexOutOfBoundsException: Range [251, 250) out of bounds for length 489

<p><a id="X7A489A5D79DA9E5C" name="X7A489A5D79DA9E5C"></a></p>

<h4>4.7 <span class="Heading">Examples</span></h4>

<p>This section provides a number of known examples that can be studied using the <strong&t;Tight 5-design rank 3 degree8,cardinality 819,and  set

<div class="example"><pre><spanclass=GAPprompt>&t<span>< class=GAPinput>(lastJordanDesignSpecialBound);/>
<span =GAPpromptgap&;</span> < G"JordanDesignAddCardinality ();/span>
&tT 3-esign   2,degree 1  4,
[ 0, 1/2 ]>
<java.lang.StringIndexOutOfBoundsException: Range [32, 5) out of bounds for length 108
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignAddCardinality(last, JordanDesignSpecialBound(last));</span>
<Tight 3-design with
[ 0, 1/2 ]>
<pan =GAPprompt"gapg;/ < ="GAPinput>(2, 4,[,/];<span>
<span class="GAPprompt">gapspan class=""gap>/ <class="APinput"(last JordanDesignSpecialBoundlast);/>
&;Tight3-esignwith rank2, egree4,  10  angleset
[ 0, 1/2 ]>
<pan class=GAPprompt"gap&;<span < class=GAPinput">ordanDesignByAngleSet ,[,/];<span
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignAddCardinality(last, JordanDesignSpecialBound(last));</span>
<Tight 3-design with rank 2, degree 8, cardinality 18, and angle set
[ 0, 1/2 ]>
<span class=GAPprompt>&;</pan <span =GAPinput>(3  [/];<spanjava.lang.StringIndexOutOfBoundsException: Index 106 out of bounds for length 106
<span class="java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
&Tight2-with  3,2,cardinality9     / &
<span class="GAPprompt">gap></pan class"">gap>/pan>< ="GAPinput>(last JordanDesignSpecialBound(last);/>
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignAddCardinality(last, JordanDesignSpecialBound(last));</span>
<Tight 3-design with rank < 11-design  rank 2,degree 23 cardinality 196560, and angle
[0, / ]&t;
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignByAngleSet(6, 2, [0,1/4]);;</span>
<lBound(last));</span>
<Tight 3-design with rank 6, degree 2, cardinality 126, and angle set
[ 0, 1span =GAPprompt>>< Gjava.lang.StringIndexOutOfBoundsException: Range [62, 60) out of bounds for length 134
<span class=[;
<span class="<span class="GAPprompt<>java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 112
&;- withrank8 degree2  64, angle set [ 1/9]gt
<s""gapgt;/ spanclass"">(5 ,[,/];<span>
<span  0, 1/3,3]gtjava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
&T -  rank5 degree4  165,
[ 0, 1/4 ]>
< class"">gapg;/>< =GAPinput>(,8,[,/,/];<span>
<span class="GAPprompt">gap>&t;Tight4-esign  with rank 2,degree 21, cardinality 275, and angle set
<Tight 5-design with rank 3, degree 8, cardinality 819, and angle set
,,/2&;
<span class="GAPprompt">gapspan class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignAddCardinality(last, JordanDesignSpecialBound(last));</span>
<java.lang.StringIndexOutOfBoundsException: Range [50, 5) out of bounds for length 134
<Tight 5-design with rank 24, degree 1, cardinality 98280, and angle set
[ 0, 1/16, 1/4 ]></pre></div>

<p>An additional icosahedron projective example is identified span ">>>span =GAPinput>(,(ast))</pan>

<div class="example"><pre><span classclass"">&t<>spanclass"APinput"(,22 [,/,/,/);/
< <span=GAPinputjava.lang.StringIndexOutOfBoundsException: Range [89, 88) out of bounds for length 134
&;5designjava.lang.StringIndexOutOfBoundsException: Range [24, 23) out of bounds for length 71
[ 0, -3/5*E(5)-2/5*E(5)^2-2/5*E(5)[0,/,/ / ]g;pre>/
2/5*E5)4 ]gt<>/>

<p>The Leech lattice short java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

<div class="example">=><java.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 134
<
<Tight 11-design with rank 2, degree 23, cardinality 196560, and angle set
[ 0, 1/4, 3/8, 1/2, 5/8, 3/4 ]>
<span class=" class="GAPpromptgt<><">java.lang.StringIndexOutOfBoundsException: Range [85, 84) out of bounds for length 108
<span class="GAPprompt">gap></span> <span class="GAPinputs class"java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 110
4  2 5 27 and
[ 1/4, 5/8 ]>2java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 84
<span classclass"java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 108
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignAddCardinality(last, JordanDesignSpecialBound(last));</span>
<Tight 5-design with rank 2, degree 6, lt;2-design with rank 4, degree 4, cardinality 36, and angle set [ 0, 1/4 ]>
[ 0, 1/3, 2/3 ]>
<">java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 113
<span class="GAPprompt">gap></spanlt2designrank,4 ,angle  19 / &t
lt4 rank java.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 73
[ /, / &;
<"java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 113
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignAddCardinality(last, JordanDesignSpecialBound(last));</span>
<ight 5design with rank 2,degree 22, cardinality 552, and angle set
[ 0, 2/5, 3/5 ]>
<class"">gap&t</span span class""JordanDesignByAngleSet,7, [,1/,/,/];<span>
<span class="GAPprompt">gap;-design with rank 4, degree 2 cardinality 60, and angle set
<Tight [0 14 1/ ]gt;
[ 0, 1/4, 1/2, 3/4 ]>
<class"java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 117
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignAddCardinality(last, JordanDesignSpecialBound(last));</span>
ltTight7design  rank,degree22,cardinality4600 andangle set
[ 0, 1/3, 1/2, 2/3 ]></pre></div>

<pSome projective designs meeting specialbound aregiven in< href"hapBib.#biBhoggar_t-designs_1982"[ Examples-11</:/>

<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput"< class="APprompt"gapgt;span>< class"GAPinput">JordanDesignByAngleSet3 ,[0];span>
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignAddCardinality(last, JordanDesignSpecialBound(last));<&t;3design with rank 3, 4  63  angleset
&3design  4  4  180,
[ 0, 1/4, 1/2 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignByAngleSet(3, 2, [0,1/3]);;</span>
<span class="[0 14 /,-1/E()-1/4E5^214E()^-/2*E(5)^,
<2-design with rank 3, degree 2, cardinality 12, and   -/*E/2*5^-/*(5^-/E5)4]g;
< class="GAPinput">(5,2,[1/];/>
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignAddCardinality(last, JordanDesignSpecialBound(last));</span>
<2-design with rank 5, degree 2, cardinality 45, and angle set [ 0, 1/4 &;5-  rank, , 32760  angle
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignByAngleSet(9, 2, [0,1/9]);;</span>
<span class="GAPprompt">gap></span> <span class[0 /2 1,1/3]gt<><>
<2-design with rank 9, degree 2, <p>Two important designs related to the <span c</>Weyl are :/>
<span class="GAPprompt">gap
<="gap&< spanclass="(,last)
<2-design with rank 28, degree 2, cardinality 4060, and anglespan="&<java.lang.StringIndexOutOfBoundsException: Range [35, 34) out of bounds for length 98
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignByAngleSet(4, 4, [0,1/ Gjava.lang.StringIndexOutOfBoundsException: Range [24, 22) out of bounds for length 134
=G>g;</pan <classG>last l);<span
<2-design with rank 4, degree 4, cardinality 36, and angle set [ 0[0 /,1,344*5-/()^2-12E5^/*(5^,
<span class="GAPprompt">gap  -/2*E(5)34E5^-34E5^-/*(,
spanclass"">gapg;</> < classGAPinput>(last, (ast)/>
<2-design with rank 4, degree 4, cardinality 64, and angle set [ 1/9, 1/3 ]>
<span class="GAPprompt">gap&t<span>< class="APinput"JordanDesignByAngleSet16, 1 019);/span>
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignAddCardinality<pan class="GAPprompt">gap></span> <span class="GAPinput">A := [ 0, 1/4, (3-Sqrt(5))/8, (3+Sqrt(5))/8 ];;</span>
& cardinality 256,and angle set [0,1/ ]gt;
<<5design with rank4 degree1 cardinality 60, and angle set
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignAddCardinality(last, JordanDesignSpecialBound(last));</span>
<3-design   -1/*E(5-/2*E()-/*(5^3-1/4*()^ ]gt</pre</div
[ 0, 1/4, 1/2 ]>
<span class="GAPprompt">gap><div classchlinkprevnextbot"&;< href="hap0">Top  Book]/>   <href".tmlcontents>C<a&bsp&<hrefchap3."[ Chapter]</a>    <a href="chap5.html">[Next Chapter]</a>   </div>
<span class="GAPprompt">gap&div class="chlinkbot"><span c"hlink1">oto Chapter:</pan>< href=."Top/a><a ="hap1html">1</><href"hap2h><a  < =chap3h">3</a>  <a ref="hap4."4<a  a =".><a  a =".tml/>  ahref=chapBib."Biba  a href".tml">nda  <div
&;3java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 68
[ 0, 1/16, 1/4 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignByAngleSet(3,<html>
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignAddCardinality(last, JordanDesignSpecialBound(last));</span>
<3-design with rank 3, degree 4, cardinality 63, and angle set
[ 0, 1/4, 1/2 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignByAngleSet(3, 4, [0,1/4,1/2,(3+Sqrt(5))/8, (3-Sqrt(5))/8]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignAddCardinality(last, JordanDesignSpecialBound(last));</span>
<5-design with rank 3, degree 4, cardinality 315, and angle set
[ 0, 1/4, 1/2, -1/2*E(5)-1/4*E(5)^2-1/4*E(5)^3-1/2*E(5)^4,
  -1/4*E(5)-1/2*E(5)^2-1/2*E(5)^3-1/4*E(5)^4 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignByAngleSet(12, 2, [0,1/3,1/4,1/12]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignAddCardinality(last, JordanDesignSpecialBound(last));</span>
<5-design with rank 12, degree 2, cardinality 32760, and angle set
[ 0, 1/12, 1/4, 1/3 ]></pre></div>

<p>Two important designs related to the <span class="Math">H_4</span> Weyl group are as follows:</p>

<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">A := [ 0, 1/4, 1/2, 3/4, (5-Sqrt(5))/8, (5+Sqrt(5))/8,</span>
<span class="GAPprompt">></span> <span class="GAPinput">(3-Sqrt(5))/8, (3+Sqrt(5))/8 ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">D := JordanDesignByAngleSet(2, 3, A);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignAddCardinality(last, JordanDesignSpecialBound(last));</span>
<11-design with rank 2, degree 3, cardinality 120, and angle set
[ 0, 1/4, 1/2, 3/4, -3/4*E(5)-1/2*E(5)^2-1/2*E(5)^3-3/4*E(5)^4,
  -1/2*E(5)-3/4*E(5)^2-3/4*E(5)^3-1/2*E(5)^4,
  -1/2*E(5)-1/4*E(5)^2-1/4*E(5)^3-1/2*E(5)^4,
  -1/4*E(5)-1/2*E(5)^2-1/2*E(5)^3-1/4*E(5)^4 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">A := [ 0, 1/4, (3-Sqrt(5))/8, (3+Sqrt(5))/8 ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">D := JordanDesignByAngleSet(4, 1, A);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignAddCardinality(last, JordanDesignSpecialBound(last));</span>
<5-design with rank 4, degree 1, cardinality 60, and angle set
[ 0, 1/4, -1/2*E(5)-1/4*E(5)^2-1/4*E(5)^3-1/2*E(5)^4,
  -1/4*E(5)-1/2*E(5)^2-1/2*E(5)^3-1/4*E(5)^4 ]></pre></div>

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