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<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a>  <a href="chap1.html">1</a>  <a href="chap2.html">2</a>  <a href="chap3.html">java.lang.StringIndexOutOfBoundsException: Index 173 out of bounds for length 61

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<><a id=" name="">/a>/p>
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<lass="ocss" /spana "chap4.html#X79A71E957D5B9755">4.1 <span class="Heading">Jacobi Polynomials</span></a>
</span>
<div
<span class=ContSS>br/<span class="nocss>n; /span>a ="chap4X872DC5F0851550401 java.lang.StringIndexOutOfBoundsException: Range [129, 127) out of bounds for length 138
<span class="ContSS<<a =" name"BD3B17F1B8391>/a><p>
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<class< classjava.lang.StringIndexOutOfBoundsException: Range [43, 42) out of bounds for length 167
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span
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<span class="ContSS"><br /><span class="nocss"> span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7C78C3A57DDC372B">4.1-2 <span class="Heading">Renormalized Jacobi />
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X82BB5F997B7B7B3A">4.2-3 <span class="java.lang.StringIndexOutOfBoundsException: Index 129 out of bounds for length 7
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X87D80CFF7F4FD7D6">4.2-4 JordanDesignQPolynomials</a></span>
<"hap4.#801BCF87B87D691">425 </a></
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<div class="ContSect"><span span class="ContSS"><br /><span>;nbsp<span><a =chtmlX82"423<span =Heading"JordanRank Degree/><a
<
<div class="ContSSBlock">
<span class="ContSS"><br /spanclass=">br ><span ="nocss>nbsp&;<span< hrefchap4XA1BCF87B87D691.5JordanDesignConnectionCoefficients>/>
<span class="ContSS"><br /><span class="nocss">  </span><a <div>/iv
</pan
<span class="ContSS"><br /><java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 7
<class="ContSS>br ><span ="nocss>&bsp&;<span>ahref".html#">4.3- JordanDesignNormalizedAnnihilatorPolynomial</
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X85E104A27EC8435A">4.3-5 JordanDesignNormalizedIndicatorCoefficients class""< /< class"nocss"&;nbsp;/pan< =".tml#X78AC6815875A2024>.-2< =H> Angle </pan</>
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 169
<class"">br/< class=n>n;nbsp<span> =chap4#"4.- </a>/panjava.lang.StringIndexOutOfBoundsException: Index 146 out of bounds for length 146
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X803F6A9986E27413">4.4 <span class="Heading< class=ContSS">br>span class=nocss>nbsp;nbsp<span>ahref"hap4.#X7923181A81406714">4.- </a></>
<
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7F730A1C7FC9D987">4.4-1 <span class="Heading">Design Cardinality class=""< /< =nocss"&;&bsp;/>a ="chap4htmlX85E1790D86685EAF36 IsJordanDesignWithPositiveIndicatorCoefficients</a></span>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html</div>/div>
</span>
<span class="ContSS"><br /><span classdiv ="ContSect"><span class"">span class=nocss" />a href"X803F6A9986E27413.4span =H>esigns AngleSet and Cardinalityspan></>
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 155
<=ContSS<br /<span class="">;nbsp<span> =ch4.- spanclass""DesignStrength</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7DBF21947AB79F1B">4.4-6 <span class="Heading">Schemes and Tight Designs</span></a>
</span>
</div></div>
<div class="ContSect"><span class="java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 7
<span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X83501A01792752CC">4.5-1 JordanDesignSubdegrees</a></span>
/<
<div class="ContSect"> < span=">&<>a =c.X7java.lang.StringIndexOutOfBoundsException: Range [105, 103) out of bounds for length 155
</span>
<div class="ContSSBlock">
<span ="ontSS"><br /><span class="nocss">  </span><a href="chap4.html#X85F7CF2D7FDF58D7">4.6-1 JordanDesignBoseMesnerAlgebra</a></span>
<class"ontSS"<br/>span class"nocss"&&;/>a href"hap4..html#X7F95AE4A7B7B5B2B">4.6-2 JordanDesignBoseMesnerIdempotentBasis</a></span>
<span class="ContSS"><br /><span class="nocss">  </span>/span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 12
java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 150
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7FC9FDCC7DD82286">4.6-6 JordanDesignSecondEigenmatrix</a></span>
<class=ContSS"><br/<span class=nocss>n; <span>a href="chap4X7FB944C182A2FBE3.7JordanDesignMultiplicities></pan>
<span class="ContSS"><br /><<div<div
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X788831797E45CE02">4.6-9 JordanDesignReducedAdjacencyMatrices</a></span>
java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 151
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7A489A5D79DA9E5C">4.7 <span class="Heading">Examples class="">br/< class="ocss"&&;<>a="chap4.#X7F95AE4A7B7B5B2B">4.2JordanDesignBoseMesnerIdempotentBasis</a<span>
</span>
</div>
</div>

<h3""><br /><spanclass"ocss"&;&bsp<span<href"hap4.#84E0CAFE83145B3F".- </>/

p spherical projective  a finite  of sphere  projective space ( < ="hapBibhbiBdelsarte_spherical_1977>DGS77</>and < ="htmlbiBhoggar_t"[]<a>for  details)  designshave special properties  interesting .The < =pkg">ALCOstrong allows  to both  andprojectivedesigns  modelling as  setsof  idempotents   simpleEuclidean algebra<p

<p>Specifically, the primitive idempotents of simple Euclidean Jordan algebras of rank <span class="Math">2</span> have the geometry of a sphere. The correspondence involves converting Euclidean inner product <span class="Math">\cos(\alpha)</span> between two unit vectors on a sphere into the corresponding Jordan innerspan =""< >class"">nbsp&bsp<span<ahrefhX7FB944C182A2FBE3.- JordanDesignMultiplicities<a<span>

<p>The tools below allow one to construct a <strong class="pkg">GAP</strong> object to represent a design and collect various computed attributes  adesignand itsparameters these does  guarantee existence   a design,althoughknownexamples andpossible  may be studied using these tools.</p>

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

<441<class=H> </span<h4

<> advantageofstudying andprojective together   ofJordanprimitive isboth sphericaland projective  canbestudied  usingJacobi polynomials, with  parameters  for the appropriate simple Euclidean Jordan algebra.</p>

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

<h5>4<div ="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7A489A5D79DA9E5C">4.7 <span class="Heading">Examples</span<span>

<div class="func"><table class="func" width="100%"><tr><td class=
<p<p>A spherical  projectivedesign a finite subset of a sphere or projective space (see <a href="chapBib.html#biBdelsarte_spherical_1977">[DGS77]</a> and <a href="chapBib.html#biBhoggar_t-designs_1982">[Hog82]</a> for more details


<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">a := Indeterminate(Rationals, "a");; java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
< ="APprompt"gap;/span> <spanclass=G"> : Indeterminate(Rationals, "b");; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">x := Indeterminate(Rationals, "x");;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">JacobiPolynomial(0,a
[ 1 ]
<
[1/a-2* /a/2b ]
<span class> advantagestudyingsphericaland designs assetsofJordanidempotents  boththespherical  projective  be studiedtogether  ,withsuitable  chosen for appropriatesimple Euclidean  ./>
1/2*a*x+1/2*b*x+1/2*a-1/2*b+x</java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

<p><a id="X7C78C3A57DDC372B" name="X7C78C3A57DDC372B"></a></p>

<h5>4.1-2 nction returns the Jacobi  < class"ab}(x)<>of class>/>java.lang.StringIndexOutOfBoundsException: Range [132, 131) out of bounds for length 519

<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="tdright">( function )</td></tr></table></div>
class> =func =100"tr">code=">#227;R_k_epsilon</>(<var ="></var var class""></var> var =Arg<var, <varclass=="Arg></>,< class=A"x/var><tdtdclass"dright"(n; )/></><tablediv
<p>These functions return polynomials of degree <var class="Arg">k</var> in the indeterminate <var class="Arg">x</var> corresponding the the renormalized Jacobi class"APprompt"gapgt<span<"APinput>(0a,b)</pan>

<p><a id="X83D7DAD082EAD26D" name="X83D7DAD082EAD26D"></a></p>

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

<p>The <strong 1/2**+/b+/*12*+/>/>

 id=X82BCF4BA84BDEE9E=X82BCF4BA84BDEE9E<a</>

<h5>4.2-h5>.- spanclass=Heading"  />/5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Range [5, 4) out of bounds for length 200
n<> =t"n;&)/>/>/java.lang.StringIndexOutOfBoundsException: Range [195, 194) out of bounds for length 201
<p>These filters determine whether an object is a Jordan design and whether the 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="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; JordanDesignByParameters</code>( <var class="Arg">rank</var>, <var class="Arg">degree</var> )</td><td class="tdright">( function )</td></tr></table></div>
<>Thisfunction constructsa  design in the manifoldof Jordanprimitiveidempotents of rank <var class="Arg">rank</var> and degree <var class="Arg">degree</var>.</p>


<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">D := JordanDesignByParameters(3,8);</span>
<designwithrank 3 and degree8gt
<span class="GAPprompt">gap></span> <
rue
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSphericalJordanDesign(D

<span classjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
true
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignByParameters(4,8);</span>
fail
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignByParameters(3,9);</span>
>

<p><a id="X82BB5F997B7B7B3A" name=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<h5>4.2-3 <span class="Heading"div="func"<table "func""%>tr<">=>#8227 /ode>>/>tdclass"dright"> ilter <td<tr</table>/div

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; JordanDesignRank</code>( <var class="Arg">>These determine   isa    the java.lang.StringIndexOutOfBoundsException: Range [102, 101) out of bounds for length 176
java.lang.StringIndexOutOfBoundsException: Range [5, 4) out of bounds for length 226
< rank of  code=code><code>  as./>


<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">D := JordanDesignByParameters(3,8);</span>
java.lang.StringIndexOutOfBoundsException: Index 105 out of bounds for length 39
<span class
[ 3, 8 ]div">pre>span ="gap;/pan< "APinput">D=(38)<>

<p><a id="span class="">&;/> < ="(;

<h5>4.2-4 java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 5

<div class
<p>Many properties of a span class="GAPprompt">gap> Gjava.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 99


div=example>>span class"GAPprompt">gapgt</span> spanclass="APinput">D := JordanDesignByParameters(3,8);</span>
<design <p>a=X82BB5F997B7B7B3A"name"X82BB5F997B7B7B3A"></a></p>
<span 
<span class="GAPprompt">gap></span> <span class="GAPinput">x := Indeterminate(Rationals, "x");;</span>
<span class="APprompt">gap></span> <span class="GAPinput">JordanDesignQPolynomials(D);</span>
function( k ) ... end
<span class=< =func> class"unc"width"00"><>td"">codeclass="">#8227 JordanDesignRank/ode var""><var>)/td< =tdright>&attribute)/>tr<table/>
[ 90, -585, 819 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func<code>( <varclass"Arg"D/> <td< =tdright>n;ttribute;)/d</>/>/div>
[ 90, -585, 819 ]</pre></div>

<p><a id="X80A1BCF87B87D691" name="java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 0

<h5>4.2-5 <design with rank 3 degree8g;

<div class[,8 ]/re>/div
<p>The <a id="X87D80CFF7F4FD7D6X87D80CFF7F4FD7D6"/>/>


<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">D := JordanDesignByParameters(3,8);</java.lang.StringIndexOutOfBoundsException: Range [35, 34) out of bounds for length 39
<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
[ [         =example<pre<class"APprompt>gap;<span>< =G"D(3,)<span
  l;design  3   8&t;
   539,5273 1/,        ,
 91191    ,1/  java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 121
=Gjava.lang.StringIndexOutOfBoundsException: Range [25, 22) out of bounds for length 114
[ 1,java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
[ 0, 1, 0, 0 ]
[ 0, 0, 1, 0 ]
[ 0, 0, 0, 1 ]</pre></div>

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

<h4>4.3 <span class="Heading">Designs with an Angle Set</span></h4>

<p>The , 585,819]

<,,819 ]<//>

<h5>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div classh5>42-5JordanDesignConnectionCoefficients<h5
<p><div="> class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ JordanDesignConnectionCoefficients</code>( <<>he connection coefficients  a  <var class=A"D/  whichlinearofcode=code"JordanDesignQPolynomials< class="rg">D/></code yield each power of the indeterminate <a href="chapBib.html#biBhoggar_t-designs_1992">[Hog92, p. 261]</a>. This attribute stores a function <code class="code">JordanDesignConnectionCoefficients(<var java.lang.StringIndexOutOfBoundsException: Index 388 out of bounds for length 0

<p><a id="X78AC6815875A2024" name="java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 21

<h5>4.3-2 <span class="Heading">Design Angle          0,0,         ,

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; java.lang.StringIndexOutOfBoundsException: Index 118 out of bounds for length 45
<iv class="><table =f" =100%">tr>td =""><ode class=func"&8227 </code> < classArg></ar )/><d class"">&bsp )/td</>/><div
<>For  design<varclass"Arg"D<var>withoutan angleset    set< class="rg"><var>as attribute<ode ="code"JordanDesignAngleSet.   < =ArgA/>must  a  of - elements <codeclass"ode"IsCyccode    <spanclass"ath"[,1)/span>. Note thatwhen< class="rg"Avar contains elements  which&; does notprovide ,inclusionin   givenabove notchecked Also  the angle  bemodifiedoncesetasan attribute of  ./>


<div class="span class="GAPprompt">></span> <span class="GAPinput">f[j][i]*JordanDesignQPolynomials(D)(i-1)))); od[1,0 , ]
<design with rank 4 and degree 4>[0 0,0 1]/pre>>
<span class="GAPprompt">gap></span> <span class="GAPinput"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
fail
<span classjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
&; with rank   degree 4&;
<span class="GAPprompt">gap></span> <span java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 0
<d with  4  4   set[1/, 13 ]gt;
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignAngleSet(D);</span>
[ 1/9,, 1/3 ]</pre>/div>

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

<h5>4.3-3 JordanDesignByAngleSet</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; java.lang.StringIndexOutOfBoundsException: Index 120 out of bounds for length 0
<p>This function constructs a new design with Jordan rank and degree given by <var class="Arg">rank</var> and <var class="Arg">degree</var>, with angle set <var class="Arg">A</var>. The same constrains on angle set <var class="Arg">A</var> given in <code class="func">JordanDesignAddAngleSet</code> (<a href="chap4.html#X78AC6815875A2024"<>This   whether anobject that satisfiescode =code>IsJordanDesign/>isequipped  angle.p


<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">D := JordanDesignByAngleSet(4, 4, [1/3, 1/9]);</span>
&;esign withrank 4,degree ,andangle [ 1/,13]&tjava.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignAngleSet(D);</span>
[/ /]/

<p><a id="X7923181A81406714" name="X7923181A81406714"></java.lang.StringIndexOutOfBoundsException: Range [0, 57) out of bounds for length 0

<h5>4.3-4 JordanDesignNormalizedAnnihilatorPolynomial</h5>

<div <design 4and degree4&;
<p>=gap; class>;/


<div class="example"java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 107
<design with rank 4, degree 4, andltd withrank 4, ,and   [19,1/ &;
<span class="GAPprompt">gap></span> <pan class=GAPprompt>></pan <class="JordanDesignAngleSetD;/panjava.lang.StringIndexOutOfBoundsException: Index 93 out of bounds for length 93
[ 1/16, -3/4, java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="GAPprompt">gap></span> <span class="GAPinput">ValuePol
0
<span class="GAPprompt">gap></span> <span class="GAPinput">ValuePol(p, 1/3);</span>
0
<span class="GAPprompt">gap></span> <span class="p>This function constructs a new design with Jordan rank and <ar =Arg">ank/>  varclass=Arg>/> anglesetvar class"rg"Avar.The constrains  set <var class="Arg">A</var> given in <code class="func">JordanDesignAddAngleSet</code> (<a href="chap4.html#X78AC6815875A2024"><span class="RefLink">4.3-2</span></a>) apply.</p>
1</pre></div>

<p><a 9 /]/>/div

<>4.-5 JordanDesignNormalizedIndicatorCoefficients/>

<div h54.- JordanDesignNormalizedAnnihilatorPolynomial</h5>
<p>The normalized indicator class=">table ="uncwidth"00%>tr>td ">code =func&8227; </code var=ArgDvar <tdtdclass""(nbspnbsp)/>>table/


<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput<p> normalized annihilator polynomial  defined for an angle set <span class="Math">A</span> as the polynomial <span class="Math">p(x)</span> of degree equal to the cardinality
<design with rank 4, degree 4, and angle set [ 1/9, 1/3 ]>
<span class="ltdesign with rank 4,degree 4  angle  [ 1/9, 1/3 ]>
[ 1/64, 7/960, 9/3520 ]
<span class=">&t;/> <span class=GAPinput"Sum([..]  -gtfi*(D(-1);/
[ 1/16, -3/4, 27/16 ]
<span class="GAPprompt /,-/4 2716]
[ 1/16, -3/4, 27/16 ]</pre></div>

<p><a id="X85E1790D86685EAF" name="java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 1

<h5>4.3-6 IsJordanDesignWithPositiveIndicatorCoefficients</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code0
<java.lang.StringIndexOutOfBoundsException: Range [44, 2) out of bounds for length 160

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

<h5>4.3-7 JordanDesignSpecialBound</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; JordanDesignSpecialBound</code>( <var class="Arg">D</var> )</td><td
<p>The special bound


<div class"xample"<pre><span ="APprompt"gapg<span  G"D: JordanDesignByAngleSet(4, 4,[/,19]);/pan>
<design with rank 4, degree
<span class
true
<> <span class=GAPinputJordanDesignSpecialBoundD;/spanjava.lang.StringIndexOutOfBoundsException: Index 97 out of bounds for length 97
64<pre<div

<p><a id="X803F6A9986E27413" name="X803F6A9986E27413"></a></p>

<h4>4.4 <span class="Heading">Designs with span class"">gapg;/ span =GAPinputSum(ist(1.3,i&;fi*JordanDesignQPolynomials()(1);/>

< java.lang.StringIndexOutOfBoundsException: Range [41, 40) out of bounds for length 318

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

<h5>4.4-1 <span class="Heading">Design Cardinality</span></h5>

<div class
< class"">=100%<class"tdleft"<classf"&8227; java.lang.StringIndexOutOfBoundsException: Range [135, 127) out of bounds for length 260
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; JordanDesignCardinality</code>( <var class="Arg">D</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Asa finite,    a . When this cardinalityisknownforan  < class="D/var>that  < class=code">sJordanDesign<var=Arg>/var><code,the  stored astheattribute <code class="code">JordanDesignCardinality(<var class="Arg">D</var>)</code>. In order to set the cardinality of a design, we can use the operation <code class="code">JordanDesignAddCardinality(<var class="Arg">D</var>, v)</code>. When <code class="code">JordanDesignAddCardinality</code> is called, the <strong class="pkg">ALCO</strong> package immediately attempts to compute <code class="func">JordanDesignStrength</code> (<a href="chap4.html#X7C4ADEEA8355A774"><span class="RefLink">4.4-5</span></a>).</p>


<div class="example"><pre><span class="GAPpromptpThe special boundof  designsatisfying <code class="code">IsJordanDesignWithPositiveIndicatorCoefficients</code> is the upper limit on the possible cardinality for the given angle set <a href="chapBib.html#biBhoggar_t-java.lang.StringIndexOutOfBoundsException: Index 226 out of bounds for length 0
<design with rank 4, degree 4, and angle set [ 1/9, 1/3 ]>
< class=GAPprompt>&<span span =GAPinput>HasJordanDesignCardinalityD;<span>
false
<spanclass=GAPprompt"gap&t<span> < class=GAPinput"IsJordanDesignWithPositiveIndicatorCoefficients()<spanjava.lang.StringIndexOutOfBoundsException: Index 120 out of bounds for length 120
<2-design java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
<="GAPinput">JordanDesignCardinality();/java.lang.StringIndexOutOfBoundsException: Index 96 out of bounds for length 96
64</pre><64/><div

<>< =" name="7A9D3F585A7CBC3<a/>

<h5>4.4-2 <span class="Heading">Designs at the Special Bound</span></h5>

<
<Asdescribed <odeclass="unc"JordanDesignSpecialBound>(ahref".html#X858B619F7E32D5EB">spanclass"efLink"4.3-</pan>/a>) we cancompute the specialboundofa using theangle. Oncecardinality is alsoknownwe  assess whether the reaches the special bound.This filteridentifieswhen a design withan angle set  cardinalityalsomeets specialbound./p>

<p><a id

<h5>4.4-=func< " ="%>tr class=tdleft>code class"">&8227 JordanDesignAddCardinalitycode>(<varclass=Arg>D/var> var ="var> )</td>tdclass="dright"(&bsp;operation&bsp)<td</r>/table><div

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code div class=func<table class"width="100%"><tr><td class="tdleft"><code class="func">‣ JordanDesignCardinality</code>( <var class="Arg">D</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<lity<var class=Arg><)/ode.<


<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<iv =e><><span class=GAPprompt>g;/ <class"APinput> :=JordanDesignByAngleSet(,, [/3,/];<>
<2-design with rank 4, degree 4, cardinality 64, and angle set [ 1/9, 1/3 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignAnnihilatorPolynomial(D);</span>
 48 108]
<spanl;2- rank,degree 44, cardinality64,andangleset[19,1/]gt;
64</pre></div>

<p><a id="X7C7C5DEF86F051F4" name="X7C7C5DEF86F051F4"></64/pre>/div>

<h54.44 />

<. ="at java.lang.StringIndexOutOfBoundsException: Range [55, 54) out of bounds for length 72
<>he    < =A><>are   <code"" < class"rg"D/ar>)/code by<code class"ode>java.lang.StringIndexOutOfBoundsException: Range [237, 235) out of bounds for length 391


<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">D := JordanDesignByAngleSet(4, 4, [1/3, 1/9]);; JordanDesignAddCardinality(D, 64);; D;</span>
<2-design with rank 4, degree 4, cardinality 64, and angle set [ 1/9, 1/3 ]>
<span class="GAPprompt">gap></span><spanclass="APinput">JordanDesignIndicatorCoefficients(D);</span>
[ 1, 7/15, 9/55 ]</pre></div>

<p><<p><a="80BE4157859BAA6A" name="X80BE4157859BAA6A"></a></p>

<<h5>.43 JordanDesignAnnihilatorPolynomial</5java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; p>The annihilator polynomial for design <var class="Arg">D</var> is defined by multiplying the <code class="code class=ArgD/ar><code by<code ="ode"JordanDesignCardinality(varclass=Arg"D/ar><code><pjava.lang.StringIndexOutOfBoundsException: Index 275 out of bounds for length 275
<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">D</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<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 value of that polynomial

<p>For a design <var class="Arg">D</var> <span class="GAPprompt">gap></span> <span class="GAPinput">ValuePol(last, 1);</span>


<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 class="GAPprompt">gap></span> <span class64<//></div>
<2-design with rank <<a id"7C7C5DEF86F051F4" name=X7C7C5DEF86F051F4"</a><p
<span class="GAPprompt">gap></span> <<h5.4- JordanDesignIndicatorCoefficients</h5>
true
<pan class="APprompt">gap></span> <span class="GAPinput">JordanDesignStrength(D);</span>
2</pre></div>

<p><a id="java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 0

<h5>4.4-6 <span class="Heading">Schemes and Tight Designs</span></h5>

<div class="func"><span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignIndicatorCoefficients(D);</span>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsAssociationSchemeJordanDesign</code></td><td class="tdright">( filter )</td></tr></table></div>
divclass"unc><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsTightJordanDesign</code></td><td class="tdright">( filter )</td></tr></table></div>
<p>These filters

<p>A design admits a <em>regular scheme</em> when <span class="div ="func>table class=func"width=100%>tr>td class="tdleft"><code class="func">‣ IsJordanDesignWithStrength</code></td><td class="tdright">( filter )</td></tr></table></div>

<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 = 2s</span> otherwise. The filter <code class="code">IsTightJordanDesign</code> returns true when the appropriate equality is satisfied for a design.</p>

<p><a id="X814E031B82F35E16" name="X814E031B82F35E16"></a></p>

<h4>4.5 <span class="Heading">Designs Admitting a Regular Scheme</span></h4>

<p><a id="X83501A01792752CC" name="X83501A01792752CC"></a></p>

<h5>4.5-1 JordanDesignSubdegrees</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; JordanDesignSubdegrees</code>( <var class="Arg">D</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For a design <var class="Arg">D</var> with cardinality and angle set that satisfies <code class="code">IsRegularSchemeJordanDesign</code>, namely <span class="Math">t \ge s - 1</span>, we can compute the regular subdegrees as described in <a href="chapBib.html#biBhoggar_t-designs_1992">[Hog92, Theorem 3.2]</a>. The subdegrees count thelt;design with rank 4 degree 4 and angle set [ 1/9, 1/3 ]>


<div class="example"><pre><span class=span class=GAPprompt"gap&<span>< class=GAPinput"JordanDesignStrength();</span>
<2-design with rank 4
<class"APprompt">&t/span><span class="APinput">JordanDesignSubdegrees(D);</span>
[ 27, 36 ]</pre></div>

<

h44. <spanclass""DesignsAdmittinganAssociationScheme/pan</

<>Whenadesign satisfies < class=Math>tge2s- 2/span>then also  an associationscheme. We can use results given in <a href="chapBib.html#biBhoggar_t-designs_1992">[Hog92]</a> to determine the parameters of the corresponding association scheme. For more details about association schemes see <a href=divclass"unc"<=f"width=100">tr< =tdleft> class"">#8227 IsTightJordanDesign/ode<td>classt>&;filternbsp;</>/</><div

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

<h5>4.6-1 JordanDesignBoseMesnerAlgebra</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; JordanDesignBoseMesnerAlgebra</code>( <var class="Arg">D</var> )</td><td java.lang.StringIndexOutOfBoundsException: Range [0, 179) out of bounds for length 0
<p>For a design that satisfies <code class="code">IsAssociationSchemeJordanDesign</code>, we can define the corresponding Bose-Mesner algebra <a href="chapBib.html#biBbannai_algebraic_2021">[BBIT21, pp. 53-57]</a>. The canonical basis for this algebra corresponds to the adjacency matrices <span class="Math">A_i</span>, with the <code class="code">s+1</code>-th basis vector corresponding to <span class="Math">A_0</span>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0


<div class="example"><pre><span class="GAPpromptjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 184 out of bounds for length 82
<span class="GAPprompt">gap></span> <span class="GAPinput">B := JordanDesignBoseMesnerAlgebra(D);</span>
<algebra of dimension 3 over Rationals>
<span class
[, A2 A3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">One(B)
A3
true</pre></div>

<p< idX83501=X83501"<<p

<h54.- JordanDesignBoseMesnerIdempotentBasis>

<div class="func"iv =func<ableclass="func" width="100%"><tr><td class="tdleft"><code class=8227 <code var""></>)/><d =tdright>&bspa&)<tr>/table>/iv
<For    <class"ode"IsAssociationSchemeJordanDesign>  candefinethe basis Bose-algebra < href=chapBibhtml#"[,pp 53-]/>/>


<div class="example"><pre><span class="GAPprompt">gap></span> <span java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 0
&;-design rank 4,degree 4,cardinality 64, and  angleset[19, 1/3 >
<span class="GAPprompt">gap></span> <spanlt-esign with rank 4  ,cardinality64,andangle [ 1/,1/ ]gtjava.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 82
<span class="GAPprompt">></span> <span class="GAPinput">od;</span>
(-5/64)*A1+(3/64)*A2+(27/64)*A3
(/)-16)+916*
(1/64)*A1>. =java.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds for length 81
< <span=GAPinput>(ordanDesignBoseMesnerIdempotentBasisIsIdempotent;spanjava.lang.StringIndexOutOfBoundsException: Index 132 out of bounds for length 132
true</pre></div>p<idXF7CF2D7FDF58D7">a>/>

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

<h5>4.6-3 JordanDesignIntersectionNumbers< class"unc"< =func"=100"<>classt"<class=func">&#n style='color: green'>8227 </>( <varclass="D/> )/td><d ="tdright>&bsp;ttribute )/></><table<div

<class=">class="unc =%>trtdclass"tdleft"<classf"&8227 </code> var class="rg><var )/dtdclass"tdright" )/td/>//>
<p>The 


<divl;-with ,java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 82
&t2  4  ,cardinality angle[ 19,1/ ]gt
<span class="GAPprompt">gap></span> <span class="GAPinput;algebra   3over >
java.lang.StringIndexOutOfBoundsException: Range [27, 22) out of bounds for length 110
<span ,A2A3 java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
truepre<

<A3

<h5>4.6-4 JordanDesignKreinNumbers</h5>

<div class="func"><table class="func" width="100%"><java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<p>The Krein numbers <span class="ivclass=func>table ="""00%"<<class"tdleft">codeclass="unc">#;JordanDesignBoseMesnerIdempotentBasis> <varclass"Arg></ar>)/d>tdclass=t>&; <td>/></></>


<classe>prespanclass=GAPpromptgapgt;/ span=GAPinputD: (4 4,[/, 1/]; (D,64;;D;</spanjava.lang.StringIndexOutOfBoundsException: Index 181 out of bounds for length 181
<2-design with rank 4, degree 4, cardinality 64, and angle set [ 1/java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<spanclass""gap>/><pan class=G> : JordanDesignKreinNumbers();<span>
<span class="GAPprompt">gap></span> <span class="GAPinput">lt-design with rank4  , 64,and angle set  / 13 &;
[[[10, 16,1]  ,20,0 ][,0 0]],
  [ [ 12, 15, 0 ], [ 15, 20, 1 ], [ 0, 1, 0 ] ], 
  [ [ 27, 0, 0 ], [ 0, 36, 0 ], [ 0, 0, 1 ] ] ]
  </pre></div>

<"=X87C0A2067F13F01A>/></>

<h5>4.6-5 span class="GAPpromptG>g;/span <span class"">ForAllJ(D) );/pan

<div java.lang.StringIndexOutOfBoundsException: Range [0, 10) out of bounds for length 0
<p>As describe in <a java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 0


<div class=" ="java.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 239
<2-design with rank 4, degree 4, cardinality 64, and angle set [ 1/9, 1/3 ]>
<=G>gapgt;/ span=GAPinput> = BasisD);<span
<span class="GAPprompt">gap></span> <span class="GAPinput">e := JordanDesignBoseMesnerIdempotentBasis(
<"APprompt>&;/><pan class""ForAll(1.],i -gt;a[]  Sum([.3,j -gt;/span>
<span class="GAPprompt">&;2- withrank4  ,cardinality64,and angle   /,13 ]java.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 82
true</pre></div>

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

<h5>4.6-6 JordanDesignSecondEigenmatrix</h5>

<div class="func"><table classtrue<pre<div>
<p>As describe in <a href


<div class="example"><prediv class"unc"<class=func =100%<<class"dleft"codeclass=func>#;JordanDesignKreinNumbers(<"><var>)/d>td =t"(nbsp&)/>/r/<div>
& with ,degree4 cardinality,and   19 /];
<java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0
<Gjava.lang.StringIndexOutOfBoundsException: Range [24, 22) out of bounds for length 116
<span class="GAPprompt">gap></span> <span class="GAPinput">ForAll([1..3]<2designwithrank4, degree4 cardinality64 andangleset[ 1/9 1/]gt
<span class="&;/span><span class=GAPinput">(1.]  - java.lang.StringIndexOutOfBoundsException: Range [113, 108) out of bounds for length 132
true
<span =GAPpromptgapg;/><pan ="APinput>(D =Inverse((D)/span>
<span class="GAPprompt">></span> <span class="GAPinput">*JordanDesignCardinality   ,16,1], ,20, 0 ] [1 ,0 ]] 
true<pre/>

<p><a id="X7FB944C182A2FBE3" name="X7FB944C182A2FBE3"></a></p[[27,0  ][0 ,0 ,[0,0   ] ]

<54.7JordanDesignMultiplicities/>

<> =X87java.lang.StringIndexOutOfBoundsException: Range [33, 27) out of bounds for length 62
<p>


<divAsjava.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 552
<2-design with rank 4, degree 4, cardinality 64, and angle set [ 1/9, 1/3 ]>
<span class="GAPprompt">java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 0
[27 36,, 1 ]/pre</iv

<p><a id="&t;2-esign with rank 4, degree 4, cardinality 64, and angle set [ 1/9, 1/3 ]>

<h5>4.6-8 <span class"GAPprompt">gap></span> <span class="GAPinput">e := JordanDesignBoseMesnerIdempotentBasis(D);;</span>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; DesignValencies</code>( <var class="Arg">D</var> )</td><td class="java.lang.StringIndexOutOfBoundsException: Index 168 out of bounds for length 111
<p>As describe in <a href="chapBib.html#java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 0


<div
, degree,cardinality,andjava.lang.StringIndexOutOfBoundsException: Range [62, 61) out of bounds for length 82
<span=< java.lang.StringIndexOutOfBoundsException: Range [151, 149) out of bounds for length 237
[ 27, 36, 1 ]</pre></div>

<p>

<</h5

< java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 244
p  <java.lang.StringIndexOutOfBoundsException: Range [25, 24) 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 class="pkg">ALCO</strong> package. The following tight projective t-designs are identified in <a href="chapBib.html#biBhoggar_t-designs_1982


< =e>prejava.lang.StringIndexOutOfBoundsException: Range [32, 31) out of bounds for length 134
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignAddCardinality(last, JordanDesignSpecialBoundjava.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 132
true
[<pan ""< "">()=(()<span
<span class="span class="GAPprompt>gt;/span> <span class="GAPinput">*JordanDesignCardinality(D);</span>
<span class="GAPprompt">gap></span> <java.lang.StringIndexOutOfBoundsException: Range [0, 44) out of bounds for length 16
<ight3- withrank ,degree 2,cardinality6,and angle set
[ 0,
<></pan <class"APinput"JordanDesignByAngleSet2 , 012)java.lang.StringIndexOutOfBoundsException: Index 108 out of bounds for length 108
<span class="GAPprompt">gap></span> <span class=" class="func"><table class="func" width="100%"><tr>=tdleft"<code ="func"&#227;JordanDesignMultiplicities</code> var=A"D< <td> =t">&;nbsp)</td</>/>/>
 3rankjava.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 71
[ 0, 1/2 ]>
<
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignAddCardinality(lastdivclass""<>spanclass"APprompt">gapg;<><class"">D: (44[/(D,64; <span
<-  rank4  ,cardinality64,andangleset /,13]gt;
[ 0, 1/2 ]>
<><span < =G>ordanDesignByAngleSet ,14);/
<span class="GAPprompt">gap></span ,,1><>
<Tight 2-design with rank 3, degree 
<class"APprompt">apg;<span spanclass"APinput"(4,,[,/);<spanjava.lang.StringIndexOutOfBoundsException: Index 108 out of bounds for length 108
<span class="GAPprompt">gap></span> <span class="GAPinput">java.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 0
<Tight 3-esign  rank4 degree 2 cardinality 40, and angle set
[ 0, 1/3 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignByAngleSet(6, 2, [0,1/4]);;</span>
span class"APprompt">></ <class=GAPinputJordanDesignAddCardinality, JordanDesignSpecialBound);<span
<Tight 3-design with rank 6, degree 2, cardinality 126, and angle set
[ 0, 1/4 ]>
<span class=="GAPprompt">gap></span> <span class="GAPinput">JordanDesignByAngleSet(8, 2, [1/9]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignAddCardinality(last, java.lang.StringIndexOutOfBoundsException: Index 99 out of bounds for length 82
&;Tight 2design with rank 8, degree 2, cardinality 64, and angle set [ 1/9 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignByAngleSet(5, 4span =GAPprompt>><span>< class=">(D);</pan>
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignAddCardinality(last, JordanDesignSpecialBound(last));</
&;ight3-esignwith  5 4  165, and angleset
[ 0, 1/4 ]>
<span h5>4.69JordanDesignReducedAdjacencyMatrices</>
<span class="GAPprompt">gap></span> <span class="GAPinput">div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ JordanDesignReducedAdjacencyMatrices</code>( <var class="Arg">D</var> )</td><td class="tdright">( attribute <p> defined  <a ="chapBibhtmlb">[CVL91, p 201</a>  reduced  matrices multiplywith the  structure  as  adjacency , which allows  asimpler construction of an algebra isomorphic to the Bose-Mesner algebra. The matrices <code class="code">JordanDesignReducedAdjacencyMatrices(<var class="Arg">D</var>)</code> are used to construct <code class="
lTightdesignwith , 8 cardinality819, and angle set
[ 0
<span
<span ""gapg;/>span class""JordanDesignAddCardinality, JordanDesignSpecialBound(last)<span>
<Tight 5-designclass"">&t/span <panclass="APinput>(last,JordanDesignSpecialBoundlast);/>
[ 0, &;ight3-esign withrank2  ,cardinality,andangleset

<p>An additional icosahedron span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignByAngleSet(2, 2, [0,1/2]);;</span>


<div class="example"><pre><span class="GAPprompt">pan class="gap&t<span><panclass=""JordanDesignByAngleSet 012);/>
<class"APprompt>&<span><span =G"JordanDesignAddCardinality, (last)<span
<Tight 5-design with ltTight 3- withrank 2  4,cardinality,andangle 
[ 0, -3/5*E(5)-2/5*E(5)^2-2/5*E(5)^3-3/5*E(5)^<pan=GAPprompt>>/>span"">(2,8,012);/>
  -2/5*E(5)-3/5*E(5)^2-3/5*E(5)^3-2/5*E(5)^4 ]></pre></div>

<p>The Leechspanclass"APprompt"gapgts> spanclass"">ordanDesignByAngleSet,2, 1/];/span>


<div class="example"><pre><span class="GAPprompt">gap></span> <span class="&t;Tight 2-esign  rank 3,degree 2  9,and angleset [14]gt;
<pan=GAPpromptgap&</><panclass="GAPinput"JordanDesignAddCardinality,JordanDesignSpecialBound)<spanjava.lang.StringIndexOutOfBoundsException: Index 134 out of bounds for length 134
&Tight11designwith2, ,cardinality196560 and angleset
[ 0, 1/4, 3/8, 1/2, 5/8, 3/4 ][0,1/]&tjava.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignByAngleSet(2, 5, [ 1/4, 5/8 ]);;</span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignAddCardinality(last, JordanDesignSpecia
<class""gap;/span><span class="APinput">JordanDesignAddCardinality(last, JordanDesignSpecialBound(last));</span>
<Tight 4-design with rank 2, degree 5, cardinality 27, and angle set
[1/4, 5/8 ]>
<">gap></pan <span class="GAPinput">JordanDesignByAngleSet(2, 6, [0,1/3,2/3]);;</span>
<span class="GAPprompt">gap&&tTight 2design  rank 8,  2,cardinality64,and angle [ &;
<Tight="APprompt>&</span> < ="APinput>JordanDesignByAngleSet5,4 0,4);/>
[ 2/ &;
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignByAngleSet(2, 21, [3/8, 7/&t;ight3designwith  5,degree ,cardinality165, andangleset
<spanspan =GAPprompt>&t</pan>spanclass=">JordanDesignByAngleSet3 8 01/1/];/span>
& d2 java.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 73
[ 3/8, 7/12 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignByAngleSet(2[ 0,1/4,1/2  ]gt
<java.lang.StringIndexOutOfBoundsException: Range [27, 22) out of bounds for length 134
<Tight 5-design with rank 2, degree 22, span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignAddCardinality(last, JordanDesignSpecialBound(last));</span>
[ 0, 2/5, 3/5 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignByAngleSet(2, 7, [0,1/4,1/2,3/4]);;</span>
<class="APprompt"gap;</span><class"GAPinput"JordanDesignAddCardinalitylast,JordanDesignSpecialBoundl);</pan
<Tight 7-design with rank
[ 0, 1/4, 1/2, 3/4 ]>
<span class=GAPprompt>apgt;/span><span class=GAPinput>JordanDesignByAngleSet(2 22, [013122/];<span>
<span class="GAPprompt">gap></<span class="GAPprompt">gap></span class"GAPinput">JordanDesignAddCardinality(last, JordanDesignSpecialBound(last));</span>
<Tight 7-design with rank 2, degree 22, <Tight 5-- with rank 2, degree 2, cardinality 12, and angle set
 0,13,12,23 ]&t</pre><div>

<p>Some projective designs   -2/5*E(5)-3/5*E(5)^2-3/5*E(5)^3-(5)4 ]gt;<pre<div>


<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignByAngleSet(4, 4, [0,1/4,1/2])
<span class=GAPprompt">ap>/span> <span class="GAPinput">JordanDesignAddCardinality(last, JordanDesignSpecialBound(last));</span>
<3-design with rank 4, degree 4, cardinality 180<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignAddCardinality(last, JordanDesignSpecialBound(last));</span>
[ 0, 1/4, 1/2 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignByAngleSet(3, 2, [0,1/3]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignAddCardinality(last, JordanDesignSpecialBound(last));</span>
<2-design with rank 3, degree 2, cardinality 12, and angle set [ 0, 1/3 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignByAngleSet(5, 2, [0,1/4]);;</span>
<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 ]>
<span">gap>/span><span class="GAPinput"JordanDesignByAngleSet(9, 2, [0,1/9]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignAddCardinality(last, JordanDesignSpecialBound(last));</span>
<2-design with rank 9, degree 2, cardinality 90, and angle set [ 0, 1/9 ]>
<pan ="APprompt">gap></span> <span class="GAPinput">JordanDesignByAngleSet(28, 2, [0,1/16]);;</span>
<span class="GAPprompt">gap><Tight 4-design with rank 2,degree 5,cardinality 27, and angleset
<2-design with rank 28, degree 2, cardinality 4060, and angle set [ 0, 1/16 ]>
<span ="APprompt">gap></span> <span class="GAPinput">JordanDesignByAngleSet(4, 4, [0,1/4]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignAddCardinality(last, JordanDesignSpecialBound(last));</span>
&java.lang.StringIndexOutOfBoundsException: Range [47, 3) out of bounds for length 80
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignByAngleSet(4, 4, [1/9,1/3]);;</span>
<span class="span class="GAPprompt>ap></span> <span class="GAPinput">JordanDesignByAngleSet(2, 21, [3/8, 7/12]);;</span>
<-design with  4 degree 4, cardinality 64 and angle set[ 1/,13]gt;
<span class="GAPprompt">gap>&;Tight 4-designwith  2,degree 21, cardinality 275, and angle set
<span class="GAPprompt">gap>[ 38, 712]gt
<<pan class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignByAngleSet(2, 22, [0,2/5,3/5]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignByAngleSetlt; 5-rank2 java.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 73
<span class="GAPprompt">gapspan =GAPprompt>g;span><=GAPinput>(2 7, 0,1/1234);/>
< 4 ,java.lang.StringIndexOutOfBoundsException: Range [48, 47) out of bounds for length 65
[0,1/,/ >java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignByAngleSet(16, 1, [span ="APprompt">gap></span> <span class="GAPinput">JordanDesignByAngleSet(2, 22, [0,1/3,1/2,2/3]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput"lt;Tight 7-design with 2  22, 4600, java.lang.StringIndexOutOfBoundsException: Range [71, 70) out of bounds for length 74
&>designs the bound  given <="htmlbiBhoggar_t-designs_1982>Hog82,Examples 111]/>:/java.lang.StringIndexOutOfBoundsException: Index 145 out of bounds for length 145
[ 0, 1/16, 1java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<pan"">&;/ span=GAPinput(3,4 0,1/4,1/2];;/>
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignAddCardinality(last, JordanDesignSpecialBound(last));</span>
&t3- with  ,degree,cardinality63,and 
[ 0, 1/4, 1/2 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignByAngleSet(3, 4, [0,1/4,1/2,(3+Sqrtlt;;-design withrank ,degree4,cardinality and angle set
<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/,12 1/*(514*()2-/*(5^-/2E^,
   -4E(5)-1/2*()212E)^-/*(5^ &t
<span class="GAPprompt">gap></span> <span class="GAPprompt">gap></span> <spanJordanDesignByAngleSet5,,[,/];<span
<span class="GAPprompt">gap></span> <span class="GAPinput">JordanDesignAddCardinality(last, JordanDesignSpecialBound(last));</span>
&t5designwith  12,degree2 cardinality32760,and set
[ ,1/2,/4 1/3 >/pre<div

<lass="Math">H_4/span Weyl group  asfollows<p


<div class="example"><prespan class=GAPprompt">gap&t;/span><span =GAPinput">JordanDesignAddCardinality(last,JordanDesignSpecialBound(last))</span>
<span class=GAPprompt"&t;/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="APprompt">gap></span> <span class="GAPinput">JordanDesignAddCardinality(last, JordanDesignSpecialBound(last));</span>
<11-design with rank 2, degree 3, cardinality 120, and angle<span class="APprompt"gap&t</> span ="APinput"JordanDesignAddCardinality(, JordanDesignSpecialBound(ast);</>
[ , 14,1/2 3/, -3/4*()12*E5)^21//*E()^-34E)4java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63
-)/*()23/*(5)312*5)^4,
  -1/2*E(5)-1/4*E(5)^2-1/4* =GAPpromptgap&t</pan span=""JordanDesignAddCardinalitylast JordanDesignSpecialBoundl))<span
  -1/4*E(5)-1/2*E(5)^2-1/2*E(5)^3-1/4*E(5)^4 ]>classGAPpromptg;/ span">(16, ,[,/];<span>
< java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 116
<span class="GAPprompt">gap></span> <span class="GAPinput">D := JordanDesignByAngleSet(4, 1, A);;</span>
<span class="GAPprompt">gap></<2-design with rank 16, degree 1,,set  / &;
-design , ,60 set
[ 0, 1/4, -1/spanjava.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 134
  /E)125^212E)^314*5^];>/>


<="chlinkprevnextbot" a=c.html>of<a&  a =chap0h#">[ontents]/>&;  nbsp;a ="chap3.html>Previousjava.lang.StringIndexOutOfBoundsException: Range [180, 179) out of bounds for length 251


<lass=chlink1>otoChapter:</pan< =chap0html><a  ahrefc.">1</  a =chap2.tml"2/><href=.tml>=c.html>4/><href"chap5.html"5/>  < href"hap6h">6<a< "hapBib.tml></>< href=chapIndh"></>  />

<hr />
<p class="foot">generated by <a href="https://www.math.rwth-aachen.delt;design with rank 16, degree 1, cardinality 2160, and angle set
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