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<><a =c.html5 =">ctonionjava.lang.StringIndexOutOfBoundsException: Range [103, 102) out of bounds for length 127
<classjava.lang.StringIndexOutOfBoundsException: Range [23, 20) out of bounds for length 188
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X86B3BD9C84639A7E">5.1-1 IsLeechLatticeGramMatrix</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X78F07E967C6779FF">5.1-2 IsGossetLatticeGramMatrix</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X824D0D267A7C0765">5.1-3 <spanvolume61<java.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for length 21
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7C043B907EC1FE89">5.1-4 0237634(8#915)<mrnumber
<span class="ContSS"><br /<rclass20.27<java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 26
</div></div>
<div class=" <mrreviewer>J. E. /rreviewer>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X781F36A17CD9FDA6">5.2-1 UnderlyingOctonionRing</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X8120692484549A5B">5.2-2 OctonionGramMatrix</a></span>
<span class="ContSS"><br /><span class=
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X80B8907778F550EE">5.2-4 LLLReducedBasisCoefficients</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X844164BC82764599 author>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7F4BF549811C33EA">5.2-6 GramMatrix</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X825D41AE7A411640">5.2-7 <span class="Heading">java.lang.StringIndexOutOfBoundsException: Index 136 out of bounds for length 13
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X79F28E887AF17FFC">5.3 <span class="Heading">Octonion Lattice Operations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html <ear>/java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 19
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X84CF857E83452B67">5.3-2 ScalarProduct</a></span>
<<pages79–8/pages>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X83B4296D7A2F59F8">5.3-4 <span class="Heading">Lattice Vector Coefficients</span></a>
</span>
</div></div>
</div>
<h3>5 <span class="Heading">Octonion Lattice Constructions</span></h3>
<p>An <em>octonion vector</em> is a tuple of octonions. Vector addition and scalar multiplication are defined in the usual way, as component-wise addition and scalar multiplication. We can define the<java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 26
<p>An <em>octonion lattice</em> is a free left ℤtype"journal>Bulletin thejava.lang.StringIndexOutOfBoundsException: Range [52, 51) out of bounds for length 64
<p>The <strong class="pkg"> Society<other
<p><a id="X86D2839985CED826" name="X86D2839985CED826"></a></p>
<>5. span class"">Gram java.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 77
<p>The Gram matrix of a lattice basis is a computed by taking the inner products of /<entry>
<p><a id="X86B3BD9C84639A7E" name="X86B3BD9C84639A7E"></a></p>
<h5>5.1-1 IsLeechLatticeGramMatrix</h5>
<div class="func"><table class="func" width="100%"><trentry id="<java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
<p>This function tests whether <var class="Arg">G</var> is a Gram matrix of the Leech lattice. In order for the function to return <code class="code">true</code>, argument <var class="Arg">G</var> must satisfy the following. First, <var class="Arg">G</var> must satisfy <code class="code">IsOrdinaryMatrix(<var class="Arg">G</var>)</code>, <code class="code">DimensionsMat(<var class="Arg">G</var
<p><a id="X78F07E967C6779FF" name="X78F07E967C6779FF"></a></p>
<h5>5.1-2 IsGossetLatticeGramMatrix</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; IsGossetLatticeGramMatrix</code>( <var class="Arg">G</var> )</td><td<ssn0020-<issn>
<p>This function tests whether <var class="Arg">G</var> is a Gram matrix of the Gosset lattice, also known as the <span class="Math">E_8</span> java.lang.StringIndexOutOfBoundsException: Index 151 out of bounds for length 26
java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
<h5><ther=fjournal">nventiones Mathematicae</other>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; MOGLeechLatticeGeneratorMatrix</code></td><td class="tdright">( global variable )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; MOGLeechLatticeGramMatrix</code></td><article><entry>
<p>The Leech lattice
<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">M := MOGLeechLatticeGeneratorMatrix;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := M*TransposedMat(M)/8;;</span>
<span class="GAPprompt"<>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsLeechLatticeGramMatrix(G);</span>
</re>/iv>
<p><a id="X7C043B907EC1FE89" name="X7C043B907EC1FE89"></a></p>
<h5>5.1-4 IsOctonionLattice</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; IsOctonionLattice</code></td><td class=" title>Three on exceptionalgroups<
<p>Asdescribed above,an lattice afree ℤby set octonion andequipped some octonionvector norm.The category codeclasscode">sOctonionLattice<ode> s subcategoryof code =code"IsFreeLeftModule/>used class="></java.lang.StringIndexOutOfBoundsException: Range [323, 322) out of bounds for length 738
<p>Let <var class="Arg">L</var> be an object that satisfies <code class="code">IsOctonionLattice(<var class="Arg">L</var>)</code>. Then <var class="Arg">L</var> is equipped with an attribute <code class="code"><var class="Arg">G</var> = OctonionGramMatrix(<var class="Arg">L</var>)</code>. The matrix <var class="Arg">G</var> is a Hermitian octonion matrix used to define the octonion lattice norm and inner product. For <var class="Arg">x</var>, <var class="Arg">y</var> octonion vectors in <var class="Arg">L</var>, the inner product of <var class="Arg">x</var> and <var class="Arg">y</var> is computed as <code class="code">Trace(<var class="Arg">x</var>*<var class="Arg">G</var>*ComplexConjugate 969</ooktitle>
<p><a id="X7FCBD0FF7D8C0C19" name="X7FCBD0FF7D8C0C19"></a></p>
<h5>5.1-5 OctonionLatticeByGenerators</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; OctonionLatticeByGenerators</code>( <var class="Arg">gens</var>[, <var class="Arg">G</var>[, <var class<uthor>
<p>This function returns a free left module that satisfies <code class="func">IsOctonionLattice</code> (<a href="chap5.html#X7C043B907EC1FE89"><span class="<java.lang.StringIndexOutOfBoundsException: Range [13, 10) out of bounds for length 13
< >Computersin java.lang.StringIndexOutOfBoundsException: Range [34, 33) out of bounds for length 75
<p>The argument <var class="Arg">G</var> is stored as the attribute <code class="code">OctonionGramMatrix</code> and is java.lang.StringIndexOutOfBoundsException: Index 124 out of bounds for length 46
<p>In the following example, we construct the octonion lattice consisting of all octavian integer triples where the norm of java.lang.StringIndexOutOfBoundsException: Index 125 out of bounds for length 41
<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">O := OctavianIntegers;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">gens := Concatenation(List(IdentityMat(3), x -> List(Basis(O), y -> x*y) pages‐39 SIAMAMS Proc,Vol.IV<pages>
[ [ (-1/2)*e1+(1/2)*e5+(1/2)*e6+(1/2)*e7, 0*e1, 0*e1 ],
[ (-1/2)*e1+(-1/2)*e2+(-1/2)*e4+(-1/2)*e7, 0*e1, 0*e1 ],
12)*+1/*3(-/)e5+-/)*,0e1, 0* ]
[ (1/2)*e1+(-1/2)*e3+(1/2)*e4+(1/2)*e5, 0*e1, 0*e1 ],
[ (-1/2)*e2+(1/2)*e3+(-1/2)*e5+(1/2)*e7, 0*e1, 0*e1 ],
[ 1/)*+(/2*e4+(1/2)*e5+(-1/2)*6 0*e1,0* ,
[ (-1/2)*e1+(-1/2)*e3+(1/2)*e4+(-1/2)*e5, 0*e1, 0*e1 ],
/2)e6(-1/)*8, *e1, 0*e1 ]
[ 0*e1, (-1/2)*e1+(1/2)*e5+(1/2)*e6+(1/2)*e7, 0*e1 ],
[ 0*e1, (-1/2)*e1+(-1/2)*e2+(-1/2)*e4+(-1/2)*e7, 0*e1 ],
[ 0*e1, (1/2)*e2+(1/2)*e3+(-1/2)*e5+(-1/2)*e7, 0*e1 ],
*1,(/)*1+-*3(/)e4+1/*5, *1 ,
[ 0*e1, (-1/2)*e2+(1/2)*e3+(-1/2)*e5+(1/2)*e7, 0*e1 ],
[ 0*e1, (1/2)*e2+(-1/2)*e4+(1/2)*e5+(-1/2)*java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
[ 0*e1, (-1/2)*e1+(-1/2)*e3+(1/2)*e4+(-1/2)*e5, 0*e1 ],
[ 0*e1, (1/2)*e1+(-1/2)*e4+(1/2)*e6+(-1/2)*e8, 0*e1 ],
[ 0*e1, 0*e1, (-1/<ame>firstJ H<first<ast><last<name>
[ 0*e1, 0*e1, (-1/2)*e1+(-1/2)*e2+(-1/2)*e4+(-1/2)*e7 ],
[ 0*e1, 0*e1, (1/2)*e2+(1/2)*e3+(-1/2)*e5+(-1/2)*e7 ],
[ 0*e1, 0*e1, (1/2)*e1+(-1/2)*e3+(1/2)*e4+(1/2)*e5 ],
[ 0*e1, 0*e1, (-1/2)*e2+(1/2)*e3+(-1/2)*e5+(1/2)*e7 ],
[ 0*e1, 0*e1, (1/2)*e2+(-1/2)*e4+(1/2)*e5+(-1/2)*e6 ],
[ 0*e1, 0*e1, (-1/2)*e1+(-1/2)*e3+(1/2)*e4+( CL/>rentzianjava.lang.StringIndexOutOfBoundsException: Range [44, 42) out of bounds for length 50
[ 0*e1, 0*e1, (1/2)*e1+(-1/2)*e4+(1/2)*e6+(-1/2)*e8 ] ]
<span class="GAPprompt">gap></span> <span<year>983<<year>
<span class="GAPprompt">gap></span> <span class="GAPinput">B := CanonicalBasis(O);;</span>
< class=GAPprompt"gap&t</span <class="GAPinput">3:= OctonionLatticeByGeneratorsgens G, B)</pan
<free left module over Integers, with 24 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">KnownAttributesOfObject(O3);</span>
[ "LeftActingDomain","Dimension","eneratorsOfLeftOperatorAdditiveGroup,
"UnderlyingOctonionRing", "UnderlyingOctonionRingBasis",
"OctonionGramMatrix", "GeneratorsAsCoefficients",
"LLLReducedBasisCoefficients" ]</pre></div>
<p><a id="X786A725B7ADE7BDE" name="X786A725B7ADE7BDE"></a></p>
<. <java.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 69
<p>An free left module <var class="Arg">L</var> that satisfies <java.lang.StringIndexOutOfBoundsException: Range [2, 1) out of bounds for length 41
<p><a id="X781F36A17CD9FDA6" name="X781F36A17CD9FDA6"></a></p>
<h5>5.2-1 UnderlyingOctonionRing</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code<thertype=oden>java.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds for length 36
<p>other ">ournal Algebra/java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
<div class="example">java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 36
<algebra-with-one of dimension 8 over Rationals></pre></div>
<p><a id="X8120692484549A5B" name="X8120692484549A5B"></a></p>
<h5>5.2-2 OctonionGramMatrix</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; OctonionGramMatrix</code>( <var class="Arg">L</var> )</td><td class="tdright">( attribute entry =">article>
<p>When <var class="Arg">L</var> satisfies <code class="java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 56
<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">OctonionGramMatrix(O3);; Display(last);</ name><first>R. A.</first><last>Parker</last></name>
[ [ (1/2)*e8, 0*e1, 0*e1 ],
[ 0*e1, (1/2)*e8, 0*e1 ],
[ 0*e1, 0*e1, (1/2)*e8 ] ]</pre></div>
<p><" name=X79113ABC7A39CFFB>/>p>
<h5>5.2-3 GeneratorsAsCoefficients</h5>
<div class="func<author>
<p>When <var class="Arg">L</var> satisfies <code class="code">IsOctonionLattice(<var class="Arg">L</var>)</code>, this attribute converts the lattice generators, <code class="code">GeneratorsOfLeftOperatorAdditiveGrouptitle>hecoveringradius the<></>echlattice<title>
<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsAsCoefficients(O3);</span>
[ [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0java.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for length 22
<ages261–290</ages>
[ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, <therjava.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 36
[ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, ,00, 0, , , 0, , , ,0, ,0, , ,0, ,0 ,0 ,0]
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0,<last><name>
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0<author
[ 0, 0, 0, 0, <Twentyforjava.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 72
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0<ssn0080-<issn>
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 ],
,0,0, ,0,0 0 0 0 ,0 ,0 ,0 ,0 ,0 ,
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0 ],
[Mathematicaland Physical Sciences<other>
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ] ]</pre></div>
<p><a id="X80B8907778F550EE" name="X80B8907778F550EE"></a></p>
<h5>5.2-4 LLLReducedBasisCoefficients</h5>
<div class="func"><table class="func" width= author>
<p>When <var class="Arg">L</var> satisfies <code class="code">IsOctonionLattice(<var class="Arg">L</var>)</code>, this attribute stores the LLL reduced basis java.lang.StringIndexOutOfBoundsException: Index 166 out of bounds for length 56
<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">LLLReducedBasisCoefficients(O3);</span>
[ [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 1,0, 0, 0,0 0, 0, 0, 0, 0, 0, 0, 0 0, 0 0, 0,0, 0, 0 ,0 0],
[ 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, <java.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for length 20
[ 0, <java.lang.StringIndexOutOfBoundsException: Range [13, 8) out of bounds for length 30
[ 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0<rclass10E05 10E25)mrclass>
[ 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, <ther"oden>BAMOADjava.lang.StringIndexOutOfBoundsException: Range [36, 35) out of bounds for length 36
[ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0
0,,0 ,0 ,0 ,0 ,
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 <Cambridgeschool>
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 type"sporsimp/
[ 0, 0, 0, 0, 0, 0, <</java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 20
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 <<R .java.lang.StringIndexOutOfBoundsException: Range [36, 35) out of bounds for length 56
0,0,0,0 ,0,0 0 0, 0, 0, 0, 0,0, 0, 0,0, 0, 1, 0, 0, 0, 0, 0 ],
stabilizers>
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0<journal>J. Algebra/ournal>
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, <java.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length 19
[ 0, 0, 0, 0 java.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length 24
<p><<rnumber>340404(9 5159<mrnumber>
<h5>5.2-5 <span class="Heading">Octonion Lattice Dimension</span></h5>
<div class="func"><table <rclass>20D05</rclass>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; Rank</code>( <var class="Arg">L</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<> satisfies <codeclass="code">IsOctonionLattice(<var class="Arg">L</var>)</code>, these attributes store the lattice dimension, also known as rank. This is calculated by evaluating <code class="code">Rank( GeneratorsAsCoefficients(<var class="Arg">L</var>) )</code>.</p>
<div class="example"><pre><span class="GAPprompt">java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
24
<span class="GAPprompt">gap></span> <span class="GAPinput">Rank(O3)<java.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for length 10
24</pre></div>
<p><a id="X7F4BF549811C33EA" name="X7F4BF549811C33EA"></a>< /java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 13
<>526GramMatrixh5
<div class="func"><table class="func" width="100%"><tr><td class="java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 23
<> java.lang.StringIndexOutOfBoundsException: Range [43, 42) out of bounds for length 506
<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">g := 2*GramMatrix(O3);;</span>
span java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 86
[ 24, 24 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">DeterminantMat(g);</span>
1</pre></div>
<p><a id="X825D41AE7A411640" name="X825D41AE7A411640"></a></p>
52< ="eading">attice Basis<span><h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; Basis</code>( <var class="Arg">L</var> )</td <ssn>021-693<issn
<div class="func"><table class="func" width="100%"><<rclass>D05</java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 26
<div class="funcjava.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
<div class="func"><table class="func"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>When
<div class="<name><first>java.lang.StringIndexOutOfBoundsException: Range [36, 35) out of bounds for length 58
true
<spanname<>last/java.lang.StringIndexOutOfBoundsException: Range [51, 48) out of bounds for length 56
[ [ (-1/2)*e1+(1/2)*e5+(1/2)*e6+(1/2)*e7, 0*e1, 0*e1 ],
[ (-1/2)*e1+(-1/2)*e2+(-1/2)*e4+(-1/2)*e7, 0*e1, 0*e1 ],
[ (1/2)*e1+(1/2)*e2+(1/2)*e4+(-1/2)*e7, 0*e1, 0*e1 ],
[ (1/2)*e2+(1/2)*e3+(-1/2)*e5+(-1/2)*e7, 0*e1, 0*e1 ],
[ (1/2)*e1+(1/2)*e3+(1/2)*e4+(-1/2)*e5, 0*e1, 0*e1 ],
[ (1/2)*e1+(1/2)*e2+(1/2)*e3+(-1/2)*e6, 0*e1, 0*e1 ],
[ (1/2)*e2+(1/2)*e4+(-1/2)*e5+(<>2java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 20
[ (1/2)*e1+(1/2)*e2+(-1/2)*e5+(-1/2)*e8, 0*e1, 0*e1 ],
[ 0*e1, (-1/2)*e1+(1/2)*e5+(1/2)*e6+(1/2)*e7, 0*e1 ],
[ 0*e1, (-1/2)*e1+(-1/2)*e2+(-1/2)*e4+(-1/2)*e7, 0*e1 ],
[ 0*e1, (1/2)*e1+(1/2)*e2+(1/2)*e4+(-1 mrreviewerN .Ajava.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
[ 0*e1, (1/2)*e2+(1/2)*e3+(-1/2)*e5+(-1/2)*e7, 0*e1 ],
[0e1 (1/2*1(1)e3+/)*4(12)*java.lang.StringIndexOutOfBoundsException: Range [47, 46) out of bounds for length 55
[ 0*e1, (1/2)*e1+(1/2)*e2+(1/2)*e3+(-1/2)other"java.lang.StringIndexOutOfBoundsException: Range [24, 23) out of bounds for length 36
[<article></entry>
[ 0*e1, (1/2)*e1+(1/2)*e2+(-1/2)*e5+(-1/2)*e8, 0*e1 ],
[ 0*e1, 0*e1, (-1/2)*e1+(1/2)*e5+(1/2)*e6+(1/2)*e7 ],
[ 0*e1, 0*e1, (-1/2)*e1+(-1/2)*e2+(-1/2)*e4+(-1/2)*e7 ],
[ 0*e1, 0*e1, (1/2)*e1+(1/2)*e2+(1/2)*e4+(-1/2)*e7 ],
[ 0*e1, 0*e1, (1/2)*e2+(1/2)*e3+(-1/2)*e5+(-1/2)*e7 ],
[ 0*e1, 0*e1, (1/2)*e1+(1/2)*e3+(1/2)*e4+(-1/2)*e5 ],
[0e1 * 1/*1+12)e2+/)e3+-/*6 ,
[ 0*e1, 0*e1, (1/2)*e2+(1/2)*e4+(-1/2)*e5+(-1/2)*e6 ],
[ 0*e1, 0*e1, (1/2)*e1+(1/2)*e2+(-1/2)*e5+(-1/2)*e8 ] ]
< class=APprompt>gap></span <class="O) ( x -t;List(b,y &t; ScalarProductO, x, ))</>
true</pre></div>
<p><a id="X79F28E887AF17FFC" name="X79F28E887AF17FFC"></a></p <ournalProc.Roy Ser </>
<h4>5.3 <><year
<p>There are some additional operations available to study octonion lattices. In the>80<java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 22
<p><a id="X87BDB89B7AAFE8AD" name="X87BDB89B7AAFE8AD"></a></p>
<h5><code>5.3-1 \in</code></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; \in</code>( <mrnumber660414(m10022a</java.lang.StringIndexOutOfBoundsException: Range [42, 41) out of bounds for length 42
<p>When <var class="Arg">L</var> satisfies <code class="code">IsOctonionLattice(<var class="Arg">Lother "coden>/>
divclass"><><panclass=GAPprompt>java.lang.StringIndexOutOfBoundsException: Range [54, 53) out of bounds for length 138
[ (1/2)*e2+(1/2)*e3+(-1/2)*e5+(-3/2)*e7, 0*e1, 0*e1 ]
<spanMathematical
true
<span class="GAPprompt">gap></span> <span class="GAPinput">\in( <thertype=porsimp>o1/ther>
true</pre></div>
<p><aid=X84CF857E83452B67 name"84F857E83452B67">/>/>
<h5>5.3-2 ScalarProduct</h5>
<div class="func"><table class="func" width="100%"><tr><td<ntryid"at72java.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds for length 29
<p>When <var class="Arg">L</var> name<irstN J/<astjava.lang.StringIndexOutOfBoundsException: Range [47, 45) out of bounds for length 59
<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">b := BasisVectors(Basis(O3));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">b[1];</span>
[ (-1/2)*e1+(1/2)*e5+(1/2)*e6+(1/2)*<itle>Conwaysgroup. java.lang.StringIndexOutOfBoundsException: Range [34, 33) out of bounds for length 56
<class=>gt;/pan><pan class=GAPinput"ramMatrix(3) =List(,x-> List(b,y-&t;ScalarProductO3, y))</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">c := LLLReducedBasisCoefficients(O3 <ear1972<year>
<span class="GAPprompt">gap></span> <span class="GAPinput">c[1];</span>
[ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">GramMatrix(O3) = List(c, x -><
true</pre></div>
<p><a id="java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 13
<h5>5.3-3 <span class="Heading">Sublattice Identification</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; IsSublattice</code>( <var class="Arg">L</var>, <var class="Arg">M</var> <journal>Arch. Math.(asel)</journal>
<div class="func"><table class="<ear>1977</year>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; \=</code>( <var class="Arg">L</var>, <var class="Arg">M</var> )</td><td class="tdright">( operation&<volume>29</volume>
<p>When <var class="Arg">L</var> and <<pages389‐391</java.lang.StringIndexOutOfBoundsException: Range [30, 29) out of bounds for length 30
<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">s := LinearCombination
<span class="GAPprompt">gap></span> <span class="GAPinput">leech_gens := List(Basis(OctavianIntegers), x -> </span>
<span class="GAPprompt">></span> <span class="GAPinput">x*[[s,s,0],[0,s,s],ComplexConjugate([s,s,s])]);;</span>
java.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 44
<span class="GAPprompt">gap></span> <span class="GAPinput">Leech := OctonionLatticeByGenerators>
<free left module over Integers, with 24 java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 0
<span class="GAPprompt">gap></span> <span class="GAPinput">IsLeechLatticeGramMatrix
true
<;</span><panclass="">IsSublattice(Leech, O3)</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSublattice(O3, Leech);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSubset(O3, Leech);</span>
true/re><div
<p><a id="X83B4296D7A2F59F8" name="X83B4296D7A2F59F8"></a></p>
<h5>5.3-4 <span class="Heading">Lattice Vector Coefficients</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; Coefficients</code>( <var class="Arg">B</var>, <var class="Arg">y</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Let <var class="Arg">B</var> satisfy <code class<olume>0<volume>
<div class="example"><java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 20
<span class="GAPprompt">gap></span> <ages>213‐227/ages>
<span class="GAPprompt">gap></span> <span class="GAPinput">rand = LinearCombination(Basis(O3), coeffs);</span><issn>0218693/
true</pre></div>
<div class="chlinkprevnextbot"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#contents">[Contents]</a> <a href="chap4.html <rclass>005</java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 26
<div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href <thertype="fjournal">Journal of Algebra</other>
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