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<         ":/www..java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
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<div class="ChapSects"><a href="chap6_mj.html#X866FC1117814B64D">6 <span class="<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.htm1- </a></span>
<div class="ContSect"><span class<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X8134BE2B8478BE8Aspan class="ContSS>br />span class="nocssspan"ContSS><br/< class"nocss"nbsp;<ahref="hap6_mj#784E1255874FCA8A"6.8 java.lang.StringIndexOutOfBoundsException: Range [134, 131) out of bounds for length 142
Functions that Generate a New Code from a Given Code
</span></a classContSS>br/>span =nocss&bsp <span>ahref".html#">6.1< =C">br/>< class"">&bsp;;/span>a =".#79912F085ACB609">.-4DualCode<a>span>
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< "">br/<panclass="css>;bsp;/>>ahref"hap6_mjA203A380BC4C796.119 StandardFormCode/<span
<pan class="ContSS"br/<class"">&;n;/pan><a="hap6_mj.html#8134BE2B8478BE8A"616 AugmentedCode></span>
<class"/iv><div>
<div=ContSectjava.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 147
<#X81CBEAFF7B9DE6EF">6.1-9  ShortenedCode</>/pan>
<span class="ContSS"><br /><span class="nocss"> &java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 12
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<span class="ContSS"><br /><span class="nocss">  <//span>/>
<span class="ContSS"><br />< =ContSS>br/<class"ocss>n;n;</>a =><>
<span=java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 7
< &bsp<span> =ctml82D18907800FE3D96.- a<span
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<java.lang.StringIndexOutOfBoundsException: Range [65, 5) out of bounds for length 147
<pan"ontSS"<r>< class"ocss"&;&bsp<span class="ContSS"><br /><span class="nocss">  </span.#X8228A1F57A29B8F46.- </></span>
<span class="ContSS"><br /><span class="nocss">  </<span class="ContSS"><br /><span class="nocss">  </span href="chap6_mj.html#X7A85F8AF8154D387">6.2-6 ExtendedDirectSumCode</aspan class=ContSS>br/>class"&;nbsp;/>href"hap6_mjXE17107686A845DB">27AmalgamatedDirectSumCode</<span>
</div></div>
<="ContSect>span ">spanclass="nocss"nbsp<span>hrefchX7964BF0081CC83526. span Code/<>
Functions that Generate a New Code from Two or More Given Codes
</span></a>
</span>
<div class="ContSSBlock">
<class="ontSS"< >span class"ocss>n;n;/<span class="ContSS"><br /><span class="nocss">  </spanc.htmlX7java.lang.StringIndexOutOfBoundsException: Range [107, 106) out of bounds for length 142
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<span class="ContSS"><br /span class=ContSS>brjava.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 134
< lassC"< /java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 141
<">6.2-5 UnionCode</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X7A85F8AF8154D387">6.2-6 ExtendedDirectSumCode</a></span>
<pan class""br/span=">;bsp<pan> ".#E17107686A845DB>27 AmalgamatedDirectSumCode<>spanjava.lang.StringIndexOutOfBoundsException: Index 149 out of bounds for length 149
<span ="ontSS>br /< =""nbsp
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X7C37D467791CE99B">6.2-9 ConstructionXCode</a></span>
<span class="ContSS
<span=">br/<span
<"ContSS>
</div></div>
</div>

<h36< =H> Codes/span>/>

<<span

<p>In some cases,

<>

<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 3141

<p><a id="X8271A4697FDA97B2" name="X8271A4697FDA97B2"></a></p>

<h4>6.1 <span class="Heading">
h5>6.1-1java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 0
<<>

<p

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

<div class="func"><tablejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<> =""<java.lang.StringIndexOutOfBoundsException: Range [42, 40) out of bounds for length 407

<p>The word length increases by <var class="Arg">i</var>. The size of the code remains the same

<span =GAPprompt"gap>/> span ="(C2 java.lang.StringIndexOutOfBoundsException: Range [95, 94) out of bounds for length 112

<div class="example"><pre>
<"java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
a linear="">apg;</>span=>2=(C1 );/java.lang.StringIndexOutOfBoundsException: Range [94, 93) out of bounds for length 94
<span class="GAPprompt">gap></span> <span classclassGAPprompt>>/ spanclass"APinputIsEquivalent , ,3  ;/java.lang.StringIndexOutOfBoundsException: Index 112 out of bounds for length 112
a linear [8,4,4]2 extended code
< =G>> spanclass""IsEquivalent java.lang.StringIndexOutOfBoundsException: Range [95, 94) out of bounds for length 112
true
<span class="GAPprompt">gap></span> <span class "">gt<span span=GAPinput: (C1 )<span>
[ 0, java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="GAPprompt
ajava.lang.StringIndexOutOfBoundsException: Range [0, 8) out of bounds for length 0
<v>

<p

<p><a java.lang.StringIndexOutOfBoundsException: Range [0, 8) out of bounds for length 0

<h5>6.1-2 PuncturedCode</h5>

<div class=div class="java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
Arg>var thelast,and the.  done simplyby  the   each codewordThis means word decreases by .Theminimum ingeneral decreaseby one<pjava.lang.StringIndexOutOfBoundsException: Index 301 out of bounds for length 301

<p>

<pPuncturing cyclicjava.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 214

<div class="example"><pre>
<span class="GAPprompt">gap></span class="GAPprompt">gap></span java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
ass">gap&<span> span =G>  BCHCode( (, 5,GF( ;/>
<span class="GAPprompt">gap>a  [15,,].5BCH< =APprompt>&/  =A"2=PuncturedCode(1)</>
a linear [linear [<class"Pprompt"ap&;pan <class""C2: PuncturedCode C1 ;/>
< class=<span class="GAPprompt&;/<panGExtendedCodeC2  =;<>
false
<span class="GAPprompt">gap><
a linear [8span="APpromptspan  class=""gapt<>< class=GAPinput"PuncturedCode ,1,23456,])<span
paceCode,5 );#Thedimensiondecreasesfrom  3<span>
a linear [3,3,1]0 punctured code
<>/>

<ass=c/ extendscodesee

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

<h5>6.1-3 EvenWeightSubcode</h5>

<java.lang.StringIndexOutOfBoundsException: Range [0, 4) out of bounds for length 0
<<5>6.1=AB67FF5621Bname=B67FF5621B//>

<>f ifjava.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 0

<div class="example"><pre>
< class="APprompt>gap><span><span class="GAPinput> =EvenWeightSubcode(( 8 (3java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
an(,3,.83.  weight subcode
<spanjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[ 0, 4, 4, 4, 4, rompt">gap&t;/> <span class"" :=EvenWeightSubcode(BCHCode(8 ,GF3  ;/>
  4, 6, 4, 6, 8, 4, 6, 8 p&t<span><
< 0, 4, 4,4, ,,4 ,6 ,  ,,,,, ,  8 4 ,java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
a linear [8,4,4]2 Reed-Muller (1,3) code over GF(2)
</pre></div>

<p> class=">xtendedCode<code> returns related code only even weights,    its dimension(java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

<p><a id="X79577EB27BE8524B" a linear[84,2Reed-- (1,3   (n 833.).8even weightjava.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38

<h5>6.1-4 PermutedCode</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code java.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 0
<p><code class="code">PermutedCode</code> returns <var class="Arg">

<div class="<p><code class="cod"PermutedCode</code returns var class="Arg"><<pre<>
<span classspan ="">ap&t
a punctured code
<span class=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
a java.lang.StringIndexOutOfBoundsException: Range [10, 8) out of bounds for length 52
<span class="GAPprompt">gap></span> <span class="GAPinput">C2 = C1;</span>
false
<panclass<panclass"">><pan>span class>2 =BCHCode15 7 GF(2))</panjava.lang.StringIndexOutOfBoundsException: Index 99 out of bounds for length 99
(,4, ,13 7, 6 812,5)
<span classdiv class=example"<>
a linear < class""gapgt<span <span classGAPinput"  ;spanjava.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
pan class="ap&<pan>< =GAPinput>C2  ;<pan>
true
</pre></div>

<p><a id="X87E5849784BC60D2" name="X87E5849784BC60D2

<h5>6.a linear [15575 <span class="GAPprompt">gap><<pan =GAPinput> =<>

<div class="func"><table class="func" width="(,4,,913
<p><code class="code

<div class="example"><pre>
<span class="h5>6.1- </h5>
[ 1, 0, 0span div class="<tableclass=f""idth=100%><r<td="dleft<class=&227;ExpurgatedCodeode> <class"></var>,< =A"></ </> class"tdright">nbsp;&;)/>/><table>/>
</pre<divjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
<span class="GAPprompt">gap></span> <span class="GAPinput">C2 := ExpurgatedCode( java.lang.StringIndexOutOfBoundsException: Range [0, 86) out of bounds for length 0
[434].11   with7word()
<span class="GAPprompt">gap></span> <<span class>&;/><spanclass="GAPinput> <<class=""></>expurgates thecode <ar Arg"Cvarg;    codewords list<class=Arg></var>.[1 0  35 105 168 280 ,435,280,168 ,35,0 0 1 ]
,0, ,14 ,0 0,1 ,0 ,0,00 ]
</pre></div>

<p>This function does not work on non-linear codes.a  [1543.].11code,expurgated word()

<p><a id="X8134BE2B8478BE8A" name="X8134BE2B8478BE8A"spanclass="gap&/span><span class=G"WeightDistribution(C2)<span

<h5>6.1-6 AugmentedCode</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; AugmentedCode</code>(<span class="GAPprompt">gap></span> <span class="GAPinput"><>div>
<ns <var class="rg"C/java.lang.StringIndexOutOfBoundsException: Range [75, 74) out of bounds for length 576

<p> command alsobe withthe <codeclass""AugmentedCode(C</>When called  list of,<codeclass="java.lang.StringIndexOutOfBoundsException: Range [170, 168) out of bounds for length 553

<div class="example"><pre>
<span class="GAPprompt">gap></span><span class="GAPprompt">gap></span> <span class="GAPinput">WeightDistribution( C2 );</span>
a linear [8,4,4]2 Reed-Muller (1,3) code over GF(2)
<span class="GAPprompt">gap></span> <span class="GAPinput">C32 := AugmentedCode
a linear  [,7,]1code augmented with 3 (s)
span class="GAPprompt">gap&</java.lang.StringIndexOutOfBoundsException: Range [0, 37) out of bounds for length 12
true
<"GAPprompt"gap&t/ spanclass"APinput> : CordaroWagnerCode6)<span>
a linear [6,2,4]2..3 Cordaro-Wagner code over GF(2java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class=G&t</pan>< ="APinput"Codeword 00,11,,])  java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
true
<a  [8,,] -Muller (,)code()
a linear [6,3,1..2]2..3 code, augmented with 1 word(s)
< =GAPprompt>apgt;/>< class=GAPinputa [,71.21,  with 3 words)
true
</pre></div>

<> function < class=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<>< ="" ="java.lang.StringIndexOutOfBoundsException: Range [0, 52) out of bounds for length 0

<>6.- RemovedElementsCode>

<div class="func"><table class="func" width="100
<>< class=spanclass=""&;/ <panclass""> : AugmentedCode<>

< class=""><pre>
<span class="GAPprompt">gap></span> <span class=java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
[ 1,span class="APprompt">gap></span> <span class="GAPinput">C32 := AugmentedCode(C31,["00000011","00000101","00010001"])>The  code, augmented with 3word()
<span classspan "">gt;/pan <spanclass"APinput> :
<span class=GAPprompt>gt;/pan spanclass
a (52013,,.15)..5 code with 35  word(s) removedGAPinput>( [,1,111  in ;/pan>
<span java.lang.StringIndexOutOfBoundsException: Range [22, 11) out of bounds for length 94
[,,0,,,168 280 435 435,280 168, 105 ,0 0,1 java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67
<span class="span =GAPprompt">apg;/java.lang.StringIndexOutOfBoundsException: Range [37, 5) out of bounds for length 119
3
</pre></div>

<tr

<p><a id="X784E1255874FCA8A" java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

<h5>6.1-8 AddedElementsCode</h5>

divclass""<class""width="%>tr> class=tdleft"> class"">8227;java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<p><code class="code">java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 0

<iv class="example>pre>
<panclass=GAPpromptle class=unc width=00%>r< "java.lang.StringIndexOutOfBoundsException: Range [74, 72) out of bounds for length 252
java.lang.StringIndexOutOfBoundsException: Range [185, 186) out of bounds for length 37
<span class="GAPprompt">gap></span> <span class="GAPinput">C2 := AddedElementsCode( C1, [ "111111" ] );</span>
a(21..)3 codewith1 word()
<2 );</span>
true
<span class="GAPprompt">gap></span> <span class
java.lang.StringIndexOutOfBoundsException: Range [0, 1) out of bounds for length 0
<span class="GAPprompt">gap></spanspanclass

</pre></div>

m  ,use<code ="ode></code> (see<span class=APprompt">>pan spanclass""C2:( L);/java.lang.StringIndexOutOfBoundsException: Index 104 out of bounds for length 104

<p><ap><odeclass=codeAddedElementsCodecode>returns code varclass=A"Cpanclass=GAPprompt>g;/pan> span GAPinput"( C2 )</span

<h5>6.1-9 ShortenedCode</h5>

<div class="func"><a cyclic [6,0,6]6 nullcode 6,66nullcode GF()
<><codeclass=code>java.lang.StringIndexOutOfBoundsException: Range [32, 30) out of bounds for length 188

<p>If <varclass="Arg"></var> is ajava.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1

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

<div class="example"><pre>
<span class="GAPprompth5>61- AddedElementsCode/h5>
a linear [15,11,a(6,,1.)java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 0
< ="APprompt>>/><class"GAPinput"C2: ( C1 ;/span>
a linear [41032shortened code
< class"APprompt"&; < =GAPinput" : ElementsCode( [1000" 1101" "0011",GF(2) );</span>
a 43,.)   unrestrictedcodeover(2)
< java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 0
2
<span class=span =G"<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 96
a (3,2,2..3)1..2 shortened code
<class=GAPprompt"&<span>panclass=A"(4);/an
[ [ 0 0 0 ], [ 1 0 1 ] ]
<="GAPprompt"gap&;/pan java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 39
a linear [31,26,3]1 Hamming (5,2)
<="GAPinput">C6:ShortenedCodeC5,[1 2  )/
a linear [28,23,3]2 shortened
span class""gapg<span spanclass="java.lang.StringIndexOutOfBoundsException: Range [0, 60) out of bounds for length 0
0
<>/>

<java.lang.StringIndexOutOfBoundsException: Range [7, 6) out of bounds for length 294

<p><a id="p>If<var class="rg>C/>  anonlinear  codeclass=codejava.lang.StringIndexOutOfBoundsException: Range [86, 85) out of bounds for length 402

<

< =func>table=func ="100"<
<java.lang.StringIndexOutOfBoundsException: Range [0, 2) out of bounds for length 0

<div class="example"><pre>
<< <var class="Arg></r>shortened taking cross section. If <var class="Arg">C</var> is a linear code, this is done by removing all codewords that start with a non-zero entry, after which the first column is cut off. If <var
a linear [5,2,3]2 Cordaro-Wagner<>This commandspan =GAPprompt>&<span>classG"C2 : ShortenedCode( C1 )<span
<java.lang.StringIndexOutOfBoundsException: Range [22, 5) out of bounds for length 96
a  [322.. code, with 1 column 1 column(s
</pre></div>

<p><code class="code">ShortenedCode</code>' shortens the code, see <code class="func">ShortenedCode</code> (<a href="chap6_mj.html#java.lang.StringIndexOutOfBoundsException: Index 133 out of bounds for length 95

<p><a id="2

<h5>6.1-11 SubCode</h5>

<div class="func"><table class="func" width="100a (,,.42user defined unrestricted code over GF(2)
<pan =""&t</> spanclass="APinput"MinimumDistance C3)<span

<div class="example[[0a (,,2.31.2 shortened code
<span span classGAPprompt>ap&;span < class=GAPinput>AsSSortedList C4 )<panclass=GAPprompt>ap;<span>< ="APinput> := HammingCode( 5, GF())<span>
a cyclic [31,11,11]7..11 BCH code, delta=[[00 0]   01]]
<span class="GAPprompt">gap></span> <span java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 34

<span class="GAPprompt">gap></span> <span classa linear [java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
[ 1java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  , 0, 0]
<span class="GAPprompt">gap></span> <span class="GAPinput">S2:= SubCode(<span class="GAPprompt">gap></span> <span class="GAPinput">OptimalityLine<span>
a linear [31,3,11]14..20java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<spaniv ="func>table  ="="00>tr>< class=t"codeclassfunc"&8227 java.lang.StringIndexOutOfBoundsException: Range [117, 115) out of bounds for length 249
[ "a linear [31,3,11]14..20 subcode of",
  "a cyclic [31,11,11]7..11 BCH code, delta=11, b=1 over GF(2)" ]
<span class="GAPprompt">gap></span> <java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 30
[ 1, 0, 0, 0, 0,< class=f"<table class=funcjava.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 0
  0 ,<>code class=ode>(  )</ode>  the  <varclass="rg"/var java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<pre<div>

< idX809376187C1525AA =X809376187C1525AA>/><p class"APprompt"gapgt<span < class="> =LengthenedCode(C1 ;/span>

<a [,22.  with1(java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54

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

<div class="example"><prediv "func>p><a id="X7982D699803ECD0F=X7982D699803ECD0F>a<
<=>g;/>span class"> =BCHCode(15,7 )<panjava.lang.StringIndexOutOfBoundsException: Index 92 out of bounds for length 92
a cyclic [15,5,7]5 BCH
<span class="GAPprompt">gap<Thisass=GAPprompt>t<> spanclass"APinput>: (311;/
a linear [8,4,4]2 a cyclic [31,11,11]7..11 BCH code, delta=
<"GAPinput">c :=Codeword  0,,,,00,10101111] );<span
<span class="GAPprompt">gap></span> <span class="GAPinput">C3a  [3111117. BCH panclass="gap&t;<> =APinput>(S1;span
a linear [7,4,3]1 residue code
</pre></div>

<p<aid="X7E92DC9581F96594 name="7E92DC9581F96594>/><p>

<h5>6.1-13 ConstructionBCode</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ConstructionBCode</code>( <var class="Arg">C</var> )</td><td class="java.lang.StringIndexOutOfBoundsException: Range [0, 176) out of bounds for length 92
<p>The function <code class="code">ConstructionBCode</code> takes a binary linear code <var class="Arg">C</var> and calculates the minimum distance0,0,0 0, 0,0 0 0,0, , pan>< ="APinput">S2:= SubCode(C, 8);</span>

< class"span class="GAPprompt">gap></span> <span class="GAPinput">WeightDistribution(S2);</span>
<class"">>/>spanclass=GAPinput> :=ReedMullerCode2,5 );/
a<pan class"GAPprompt"gap&;<span><pan class=GAPinput"WeightDistribution(S2);</span>
<span class="GAPprompt">gap></span[1 0,0 0,0, 0  0 0,,0 ,0 0]
a ajava.lang.StringIndexOutOfBoundsException: Range [29, 27) out of bounds for length 62
<span class="GAPprompt">gap&
rec(n: 24   9 q 2 :   ,
  construction := [ [ Operation> function<class=code>ResidueCode
       "UUVCode" ] [[ [Operation "DualCode" java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
              java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 0
                  [[Operation C"][6 ]]]]
          [ [ Operation "<span class="GAPprompt">gap>><class=G"C2=ResidueCode  );<span
   := ["b(,9)8 u u+  C1 andC2:"
      "Lb(12,7)=4, u
      "Lb(6,5)=2div class="example"><pre>
      -Wagnerclass=GAPprompt>&<>< =GAPinput"1:BCHCode 15  );pan>
   : 8
  upperBoundExplanation [431residue
                             contradict:", "Ub
<pre<>

<p<id"799" java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 0

<6.-4DualCode<java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class=">/>
<p><code class="code">java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 0

<> code is code,ven  varclass"><var>is non-./>

<>fa var =A"Cvar> isequaltoitsdualcode,itisjava.lang.StringIndexOutOfBoundsException: Range [79, 77) out of bounds for length 101

<div span classGAPprompt>><span span =G" =ReedMullerCode 2  ;/>
<span class="GAPprompt"  construction [[Operation"UVCode" ],
a linear [8,4,4]2 Reed-Mullerspanclass="GAPprompt>&t;/span      [  Operation "" ]    Operation  DualCode ,
<span class="GAPprompt">gap></span> <span class="span class="GAPprompt">gap></span> <span class="GAPinput(24,9 2 ; #so C2 java.lang.StringIndexOutOfBoundsException: Range [18, 1) out of bounds for length 68
java.lang.StringIndexOutOfBoundsException: Range [10, 1) out of bounds for length 78
<></pan <spanclass"">  RD;/>
true
<span class="GAPprompt">gap></span> <span           "UVCode lowerBoundExplanation :  L(249)8  +  of  and "
a cyclic [7,7,1]0 whole space code over GF(4)
<span class="GAPprompt">gap repetition code",
rue      "(2=,Cordaroagner code" "(122=,Cordaro- code ,
</pre</div>

<p><a java.lang.StringIndexOutOfBoundsException: Range [29, 5) out of bounds for length 75

<h5>6.1-15 ConversionFieldCode</h5>"(65=2,dual  the repetition",

<div class="func"><java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 19
<p><code class="code">ConversionFieldCode</code> returns the code obtained from <var java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p>See</e<div>

<divclass"xample"pre>
<span java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 0
a cyclic [4,1,4]3 repetition code overjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="GAPprompt">gap></span> <span class="GAPinput">R2 := ConversionFieldCode( R );</span>
linear[8,,]..
<span class"">&;/span><class""Size  )= Size R2 )</>
true
sGAPprompt>gt>class=GAPinput>(  );<span
[ [ Z(2)^0, Z(2)^0, Z(2)^0, Z(2)^0 ] ]
<span class="GAPprompt">gap></span> <span java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 26
[ [ Z(2)^0, 0*Z(2), Z(2span class=GAPprompt>gap>/span> span class="GAPinput">RD := DualCode( R );</span>
  [0Z) (2  844] 1,  (2
</pre></div>

<a=java.lang.StringIndexOutOfBoundsException: Range [29, 27) out of bounds for length 62

<h5>6.1-16 TraceCode</  >gt<classGjava.lang.StringIndexOutOfBoundsException: Range [62, 60) out of bounds for length 106

< java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 216
< class="Arg">F</var> (<ArgF/>the`asefield)/>

<p>Output: The linear code generated by <span class="SimpleMath">\(Tr_{E/F}(c)\)</span>, for all <span class="SimpleMath">\(c \in C\)</span>.</p>

<p><code class="code">TraceCode</code> returns the image of the code java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 0

< seem tobeeasytorelatedthe parametersof the tracecode tothe originaljava.lang.StringIndexOutOfBoundsException: Index 107 out of bounds for length 4

< class=java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 0
<span class="GAPprompt">gap></span> <span java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 0
a  [,,?]randomlygenerated codeover (java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
5
<span class="GAPprompt">gap></span> <span class="<p><code class="code">ConversionFieldCode</code> returns the var =Arg"C/ java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
linear [0711.user unrestricted overGF()
1
</pre></div>

<p><a id="X8799F4BF81B0842B" name="div ="xample>pre

<h5>6.1-17 CosetCode</h5>

<java.lang.StringIndexOutOfBoundsException: Range [24, 4) out of bounds for length 242
<>codeclass"code"CosetCode</< classclass"APprompt">apgt<span>span =">Size  )=Size(  );/span>

<p>Generating a coset is also possible by simply adding the word <var class="Arg">w</var> to <var class="Arg">C[[Z(^0, Za 8,434code convertedto basefield GF()

<div class=example"<>
<span class="GAPprompt"<panclass"APprompt"gap&java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 12
3,2)java.lang.StringIndexOutOfBoundsException: Range [37, 36) out of bounds for length 47
<span class="GAPprompt">gap></span> java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
false
<span class="GAPprompt">gap[[Z)^,0<class==unc" ="func"width=%"tr< class=""<class=func>#8227; raceCode/code(< =Arg>C<var>)td<<: <arclass"">C/>isa  code  over an <span =S
a 716,) coset
<span class="GAPprompt">gap></span> <span class="GAPinput">List(AsSSortedList(C), el-> Syndrome(H, el));  # All elements of the coset have the same syndrome in Hjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[    ,[ 1 11 ][111,[111] 1 [111 ,
  [ 11  ] [1  1 ,[1   ,[1  1]  <pan=GAPprompt"&;<>span =GAPinput">:(104G(4) MinimumDistance)<span
  [ 1 1 1 ], [ 1 1<panclass=GAPprompt">gap></span> <span class="GAPinput">trC:=TraceCode(C,GF(2)); MinimumDistance(trC);</span>
</<>

<p><a id="X873EA5EE85699832" name="

<h5>6.1-18 ConstantWeightSubcode</h5>

<div class="func"><table class="func" width="100%"><tr><td classjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p><code class="code">span class="GAPprompt">gap></span>GAPinput>:RandomLinearCode(10,4,GF(4));

<p>This command also can be called with the syntax <code class="code">java.lang.StringIndexOutOfBoundsException: Index 87 out of bounds for length 1

<p><code class="code">ConstantWeightSubcode</code> first checks if Leonejava.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 115

<java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0
<span class="GAPprompt">gap></span<span
[ 1, 0, 0, 0, 0, 0, 112, 0,java.lang.StringIndexOutOfBoundsException: Range [0, 1) out of bounds for length 0
<span class="GAPprompt">gap></span> span class="> : H c;/>
16306.16).8codejava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 22
<span class="GAPprompt"gapgt;/pan>< =GAPinput>(C)<span>
p> ais  bysimply [ 1 1 1 ], [ 1 1 1 ], [ 1 1 1 ], [ 1 1 1 ], [ 1 1 1 ], [ 1 1
<span class="GAPpromptd "xample"<>
[ 1, 0, 0, 0, 0, 0, 264, 0, 0, 440, 0, 0, 24 ]
<>>/><java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 100
ith java.lang.StringIndexOutOfBoundsException: Range [44, 40) out of bounds for length 52
<span class="GAPprompt">gap&<h5>6.1-18 ConstantWeightSubco
,,,, 0 ,
<>>

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

<

<div class="func"><table class="<
<java.lang.StringIndexOutOfBoundsException: Range [38, 2) out of bounds for length 274

<p>If <var class="Arg">C</var> is a linear code, the generator matrix and parity check matrix are put in standard form. The generator matrix then has an identity

<p>If <var[ 1,0,0 0,0 ,112,0,30 0,112 0 ,0,0,0  java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58

<div class"">pre
<pan "APprompt"  0,0 00 , 0 ,,0,,0, 0 ,,0
<span "GAPprompt">gt;span>spanjava.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 127
a linear [4,[1 0, 0 ,0 0,264 0 0 ,,0,24 ]
< class="gap>/pan><pan =GAPinput"Codeword (G ;/>
[ [ 0 1 0 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="GAPprompt">gap></span> <span class="GAPinput">Codeword( GeneratorMat( java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
[ [10  1], [0 1 0spanclass=APprompt>>/ span ="APinput"N: NordstromRobinsonCode(); (N;/java.lang.StringIndexOutOfBoundsException: Index 122 out of bounds for length 122
</pre></div>

<p><a id="X7EF49A257D6DB53B" name="X7EF49A257D6DB53B"></a></p>

<61- <>

<div class="func"><table class="func" width="100%java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<>class"ode"PiecewiseConstantCode>returns  with lengthspanclass"SimpleMath"(n

<p>A constraint is a list of integers, and a word <span class="SimpleMath">\(cjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p>An example might make things clearer:</p>

<divclass=example"><pre>
<class=GAPprompt>g;span span class=GAPinput">span class="GAPprompt">></span> <span class="GAPinput">"random form(2)<span>
     [ [ 0, 0 ], [ 0a linear[,2,1..2].2random formcode  class"GAPinput>C<span>
the C  programs are compiled,so  Leonsbinary...
the C code programs are compiled, so using java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 0
the C code programs are compiled, so using Leon's binary....
the C code programs are compiled, so using Leon's binary....
a(,7,1..)..5 piecewise constant code java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="GAPprompt">[ [ 1 0 0 1 ], [ 0 1 0 1 ]
[ [ 0 0 0 0 0 ], [ 0 0 1 1 1 ],
  [1  0 1  ],[ 11 10 1 ], [1  1 10 ]]
</pre></div>

<>The firstconstraint is satisfiedbycodeword1,the second  2   by codewords3 and4 and thefourthcodewords 5 6and 7<pjava.lang.StringIndexOutOfBoundsException: Index 153 out of bounds for length 153

<p>aid="7964F0081CC8352"name="X7964BF0081CC8352

<h4>6.2 <span class="Heading">
Functions that Generate a New Code from Two or More<iv class=="xample"<pre
/<

<p><a[[0,0 ],[0 3],[1 0 ] [2 2 ]] [,,1.].2 randomform  over GF(2)

<<5>

<class=func Cjava.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
<p><code class="code">DirectSumCode</code> returns the direct sum of codescode programsare compiled, sopre<div

<p>If both <var class="Arg<pan  class=="APprompt"gapgt;/span> span class="GAPinput

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

<iv class="xample"><pre>
<="GAPinput">C1: (  1,0] [45,GF7));/spanjava.lang.StringIndexOutOfBoundsException: Index 116 out of bounds for length 116
<span p>The first java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 0
<panclass"java.lang.StringIndexOutOfBoundsException: Range [0, 22) out of bounds for length 0
<span class="GAPprompt">gap></span> <span class="GAPinputjava.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 0
[ 3 ], [ 4 5 0 0 0],[ 4 5 33 3 ]java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
spanGAPprompt>t</><=APinput>D  +;    sum/>
true
</pre></div>

<p><a id="X86E9D6DE7F1A07E6" name="X86E9D6DE7F1A07E6"></a>

<h5>6.2-t C code java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 0

<div class="func"><table class="func" width=p<code class="code"DirectSumCode/code> returnsthedirect sum of codes <var class="the C  programs are , so usingLeon'sbinary...
<>codeclasscode>UVCode/>  the -called <span class="SimpleMath">\((u\|u+v)\)</span> construction applied to <var class="Arg">C1</var> and <var class="Arg">C2</var>. The 000   ],  0111 ]    0 0  ,[10

<p>If both <var

<p>The function <code class="code">DirectSumCode</code

<div Functions that Generate  New from  or  GivenCodes
<span class="GAPprompt">gap></span> <span <span class="GAPprompt
a cyclic [,3,]1 even weight subcode
<span class="GAPprompt<< class="gap&t</><span =GAPinput>AsSSortedListD;/>
a cyclic [4,1,4]2 repetition code <ivclass="< =" =100%>trtd<span class="GAPprompt">gap></span> <span class="GAPinput">D  =  sum</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">R := UUVCode
a linear<6- UUVCode/5java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
<span class=GAPprompt>>/ <java.lang.StringIndexOutOfBoundsException: Range [43, 4) out of bounds for length 242
true
</pre></div>

<p><a id="X7BFBBA5784C293C1" name="X7BFBBA5784C293C1"></a></p>

<h5>6.2-3 DirectProductCode</h5>

<div class="func"><table class="func
<p><code class="code">DirectProductCode</divclass=="xample"span =GAPprompt>><span><class="APinput" : ElementsCode[[1,] 4]] 7));/>

<p>If <var class="Arg">Ci</var> is a <span a cyclic [4,3,] even weight subcode

<div class="example"><span class"APprompt"g;/ span =>(D</java.lang.StringIndexOutOfBoundsException: Index 84 out of bounds for length 44
<span class="GAPprompt" [1     ] [  03 ap&t<span<  class=""R =(,C2>
a <span cl="GAPprompt"gap;/> <spanclass=GAPinput">  C1 +C2;   # addition = direct sum</span>
<span class="GAPprompt">gap></span> <span class
[,32<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 93
<span class="GAPprompt">gap></span> <span class=true
a linear [80,20,12]20..45 direct product
</pre></div>

<java.lang.StringIndexOutOfBoundsException: Range [0, 2) out of bounds for length 0

<h5>6.2-4 IntersectionCode</h5>

<div class="func"><table class="func" width="100%"><trp<codejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p><code java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 0

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput"spanclass="APprompt>gap&<span class="GAPprompt">gap> span class""C2: (4 GF();/pan>
[ a cyclic [7,7,1]0 enumerated code over GF(2),
    [,,.2]enumerated codeover GF2)
   cyclic [7,3,1..4]2..3 enumerated code over GF(2),
   over GF()
  a cyclic [7,3,1..4]2..3 enumerated code over GF(2)a  [0,20,12]..45 direct  code
  a cyclic [7,4,1..3]1 enumerated code over GF(2),
  a cyclic [7,1,7]3 div class="func"><table class"
  a cyclic [7,4,1..3]1 enumerated code over GF(2) <
<span class="GAPprompt">gap></span> <span class="GAPinput">IntersectionCode(C[6],<p><code class=="ample><pre>
true
</pre></div>

<p>The <em<div class=example>pre

<p><a id="X8228A1F57A29B8F4" name="X8228A1F57A29B8F4"></a<a cyclic [,,..]2.3 enumeratedaGF)

<h5>6.2-5 UnionCode</h5>

<div class="funcspan class=G>&;/>< =GAPinput> : DirectProductCode(,L2)<span>
<p><a linear [80,20,12]20..45 direct product code

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := GeneratorMatCode([[1,0,1],[0,1,1<java.lang.StringIndexOutOfBoundsException: Range [0, 5) out of bounds for length 0
a linear [3paid"8228 ="A1F57A29B8F4/>
<span class="GAPprompt">gap></span> <span java.lang.StringIndexOutOfBoundsException: Range [0, 50) out of bounds for length 24
linear [
<span class="GAPprompt">gap></span> <span class="p><code class="code">UnionCode</code> returns the union=example"pre
linear 331]union xample>pre
>A" =java.lang.StringIndexOutOfBoundsException: Range [76, 75) out of bounds for length 99
false
<  a cyclic .42.3code(2
false
<class"APprompt">&t/span span class"APinput"c ;span
rue
</pre></div>

<p><a id="X7A85F8AF8154D387" name="X7A85F8AF8154D387"></a></p>

<h5>6.2< class"GAPprompt">a   [,4131enumerated  2)

<divclass=func"< class"true
<p>The extended direct sum construction is java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 0

<p>java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 4

<p class="center">\[
\Psi( L, B ) = \max_{u \in F_2^{n_L}} \frac{1}{2^{k_B}}
               \um_v\ B} {r d  v )java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
\<p

<p>However, this computation will not idth"100"

<p>If <span class="SimpleMath">\(L \subseteq B\)</span>, and <span java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">c\si La [2,21 codedefined generator  overGF(java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
a linear [7,4,3]1 Hamming (3\<pjava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
< class"GAPprompt>&t</span> < class="GAPinput"d:=( 7,GF2))<span>
a =GAPprompt>&;</pan <java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 90
< =GAPprompt"&;</><java.lang.StringIndexOutOfBoundsException: Range [0, 44) out of bounds for length 0
a [2115,1..32 3fold extendeddirect sum code
</pre></div>

<p><a id="X7E17107686A845DB" name="span class="GAPprompt">gap></span> <span class="GAPinput">c := Codeword("010");; c

<>62- </h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><codefalse
<p><code class="code">AmalgamatedDirectSumCode</code> returns the amalgamated direct<panclass"GAPprompt">gap></span> <span class="GAPinputjava.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 54

<>aid"7java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 0

<p>If <varh5.2-v class"unc"table =100>rtd class=dleft> class="func">‣ AmalgamatedDirectSumCode</code>( <var class="Arg">c1</var>, <var class="Arg">c2</var>[, <var class="Arg">check</var>] )</td><td class="tdright">( function&

<div class="example"><pre>
< class="GAPprompt">gap></span> <span class="GAPinput">c := HammingCode( 3, GF(2) );</span>
linear431()code GF
<span class="GAPprompt">gap></spanP(,
alinear[6,58 ReedMuller(,)odeover2)
<span class="GAPprompt">gap></span> <span class="GAPinput">e := DirectSumCode( c, d );</span>
a linear [23,9divclass=However  java.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds for length 108
< class="APprompt"gap>
<span class="GAPprompt">gap></span> <span class="java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 47
<span class="GAPprompt">gap></span> <span class="GAPinput">CoveringRadius( f );; </div class="xample>pan ="GAPprompt"gapgt</ span class=G"d:= ReedMullerCode 1,4 )/pan
<span =APprompt>gap&t</pan><span class="GAPinput">f;</span>
a linear [22,8,3]7 span class="GAPprompt">gap>> <spanGAPinput"e := DirectSumCode( c, d );</span>
</re</iv>

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

<h562-8 BlockwiseDirectSumCode</h5>

<div class="func"><table class="func<pan class="GAPprompt">gap></span> <span class="GAPinput">f;</span>
<p><code class="code">BlockwiseDirectSumCode</code> returns a subcode of the direct sum of/pre<div

<p>In the first case, the blockwise direct sum code isajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<pclass=center\java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
ds
<h5>6.2-

<p>where<spanclass="SimpleMath">\< class="enter"\java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20

<pIn secondcase,itis definedas]/java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6

<p class="center">\[
java.lang.StringIndexOutOfBoundsException: Range [21, 3) out of bounds for length 62
\]</p>

<p>The length r>\[

<div class="example"><pre>
<span class="p>The length of the   <>&< java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 98
<span classdivclass"xample"<>
<="gapjava.lang.StringIndexOutOfBoundsException: Range [30, 5) out of bounds for length 97
<span class="GAPprompt">></span> <span class="GAPinput">C2, [[ 0,0,0,0,0,0 ],[ 1,0,1,0,1,0 ]] );</spanclass"APprompt>&;<> span class=GAPinput">e := DirectSumCode( c, d );</spanspan class=G"gap;/span span class="GAPinput">C2 := EvenWeightSubcode( WholeSpaceCode( 6, GF(2) ) );;</a linear [23,,3]7 direct sum code
Gjava.lang.StringIndexOutOfBoundsException: Range [24, 22) out of bounds for length 71
</pre></div>

<> idX7C37D467791CE99B=X7C37D467791CE99B</a/>

<h5>6.2-9 ConstructionXCode</h5>

<div class="func"><table class="func" java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 37
<  of <java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 1230

<p X, refer <hrefchapBib_mjbiBSloane72>[RC72]/>./>

< class<vclass="tableclass=f"=100"<>td ="><code="unc>#8227 ConstructionXCode/> <ar class==A"C/var>,<var class="rg"Avar )/td<td classtdright> function&
<span class="GAPprompt">gap></span> <span class="<>onsider a list of class="impleMath>(java.lang.StringIndexOutOfBoundsException: Range [0, 50) out of bounds for length 6
acyclic[127,2943]1.59 BCHcode,delta43,b1 overGF()
<class"GAPprompt"gap&t</pan span=G"2: (127,47;/span>
a cyclic [127,22,47..51]36..63 BCH code, java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 0
<span class="GAPprompt">gap></java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 20
a cyclic [127,15,55]41..62 BCH code, delta=55, b=1 over GF(2)
<span class="GAPprompt">gap></span> <span class="GAPinput">G1 := ShallowCopy( GeneratorMat(C2) );;</span>
<">&t;/>< class=GAPinputG1,[GeneratorMatC1)23] ]);/>
<span  127,,]41.  code =55 1over GF2java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
a linear [127,23,1..43]35..63 code defined by generator matrix over GF(2)
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimumDistance(span class="GAPprompt">gap></span> <span class="GAPinput">C2 := EvenWeightSubcode6,GF(2)) );</java.lang.StringIndexOutOfBoundsException: Index 123 out of bounds for length 103
43
<span class="GAPprompt">gap></span> <span class="GAPinput">C := [ C1, C2, C3 ];</span>
[ a linear [127,23,43]35..63 code defined
  a cyclic [127,22,47..51]36..63/><>
  a cyclic [127,X7C37D467791CE99Bname"X7C37D467791CE99B"<a</>
<span class="GAPprompt">gap></span> <span class="GAPinput">java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 32
true
< java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 90
java.lang.StringIndexOutOfBoundsException: Range [104, 4) out of bounds for length 4
<span class="GAPprompt">gap></span> <span class="GAPinput">A := [ RepetitionCode(4, GF(2)), java.lang.StringIndexOutOfBoundsException: Index 106 out of bounds for length 106
  java.lang.StringIndexOutOfBoundsException: Range [12, 10) out of bounds for length 48
  a
< "": (,;/
a linear [148,23,53]43..74 Construction X code
<span class="GAPprompt">gap></span> <span class="GAPinput">History(CX);</span>
[ "a linear [148,23,53]43..74 Construction X code of",
  "Base codes: [a cyclic [127,15,55]41..62 BCH code, delta=55, b=1 over GF(2)\
, a cyclic [a  [148,23,,5343..74  cyclic [127,22,47..51]36..63 BCH delta47,b1over(2)
[127,23,3]5.63 code definedbygenerator matrix overGF() ],
  "uxiliary codes: [ a cyclic [4,1,4]2 repetition code over GF(2), a cyclic [\
17,8,6]3..6 even weight subcode ]" ]
<pre>/div>

<p>< id"X7B50943B8014134F" name="X7B50943B8014134F"></a></p>

<h5</h5>

<div class"func"<table class="func"width="100%">trtdclass==tdleft"><  "Auxiliary codes [acyclic [4,4]2 repetition code over GF(2), a cyclic [\
<p>/pre></div

<>java.lang.StringIndexOutOfBoundsException: Range [7, 6) out of bounds for length 496

<ul>
<li><p><span class="h5>6.2-10 ConstructionXXCode>

<li
<li><p><a cyclic 127,,47..51]36.63 BCH , delta=47, b1 over GF(),

</li>
<li><p><p>The codewclassjava.lang.StringIndexOutOfBoundsException: Range [45, 43) out of bounds for length 496

</li>
<li>=java.lang.StringIndexOutOfBoundsException: Range [32, 30) out of bounds for length 88

</li>
</ul>
<p>For more information on Construction XX, refer to <a href="chapBib_mj.html#biBAlltop84">[All84]</a>.</p>

<>java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 123
<span class="GAPprompt">gap></span> <span class="GAPinput">a := PrimitiveRoot(GF(32));</
Z(2^5)
<java.lang.StringIndexOutOfBoundsException: Range [22, 5) out of bounds for length 107
x_1+Z(2)^0
<span class="GAPprompt">gap></span> <span class="GAPinput">f1 :=<i<><span class= 127,,.]63 ,47 =1 GF,ajava.lang.StringIndexOutOfBoundsException: Range [79, 77) out of bounds for length 79
x_1^5
<span class="GAPprompt">gap></span> <span class="GAPinput">f5 := MinimalPolynomial( GF(2), a^5 );</span>
x_1^5+x_1^4+x_1^2+x_1+Z(2)^0
<span class="GAPprompt">gap></span> <span class="GAPinput">C2 :=></div>
a cyclic [31,6,15]10..13 code defined by check polynomial over <>a <p>For m informationon Construction, referto a =chapBib_mjhtmlbiBAlltop84>All84]</></
<<=A"java.lang.StringIndexOutOfBoundsException: Range [65, 64) out of bounds for length 146
a cyclic [31,6,15]10..13 code definedspanjava.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 96
<"java.lang.StringIndexOutOfBoundsException: Range [24, 22) out of bounds for length 129
a linear [31,11,11]7..11 union code of
U: a cyclic316,]0.13codejava.lang.StringIndexOutOfBoundsException: Range [41, 40) out of bounds for length 71
V: a cyclic [31,6,15]10..13 code
<""&;/ <classGAPinput> =BestKnownLinearCode( 10, 5, GF(2) );</span>
a linear [10,5,4]2..4 shortened span class="GAPprompt">gap></spa class=GAPinput">f0 : ( 2) java.lang.StringIndexOutOfBoundsException: Index 95 out of bounds for length 10
<lip<span class"impleMath\  ;\ \; 0 \ \<class="><span<class=G> : ( (2,a5)<java.lang.StringIndexOutOfBoundsException: Index 107 out of bounds for length 107
a cyclic [6,5,2]1 dual codespan="java.lang.StringIndexOutOfBoundsException: Range [22, 18) out of bounds for length 140
<span class="GAPprompt">gap></span> <span class="GAPinput">CXX:= java.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 68
a li<><pan class="impleMath"\(v;\|;0\ \| \<spanif=G"gap&;/span>span class=APinput"3=CheckPolCodef0*f5 31 GF( ; MinimumDistance3;DisplayC3;/span>
<span class="GAPpromptspan class="APprompt>&</> <span class=>pspan =SimpleMath>( \ |\  \\ \ )<span />
17
<span class="GAPprompt">gap></java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
[ "a linear [47,11,17]13..23 Construction XX code of
  "C1: a cyclic  class=example"><pre>
  G"g;/span<java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 96
  "C3: a cyclic [31,6,15]10..13 code span class==GAPprompt"gap&</pan> <pan class"GAPinput" : java.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 36
  "A1: a linear [10,5,4]2..4 shortened code",
  ": a java.lang.StringIndexOutOfBoundsException: Range [16, 15) out of bounds for length 37
/><div

<"<a></p>

<h5>6.2-11 BZCode</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><pan ="GAPprompt" [,,].13 by   GF
<p>Given a set of outer codes of the same length <span class="SimpleMath">\(java.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 45

<>Note   3 ajava.lang.StringIndexOutOfBoundsException: Range [16, 15) out of bounds for length 78

<>or moreinformation on BlokhZyablov multilevelconcatenated code, refer to <a java.lang.StringIndexOutOfBoundsException: Index 87 out of bounds for length 37

p< =java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 62

<alinear[7,1,.17]3.23Construction

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><java.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 0
<p>This function is the same as <code class="code">BZCode</code>, except this version is faster as it does not estimate the covering radius [ajava.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 55

<div class="example">":cyclic 3161510.codejava.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 78
<span class="APprompt"gap&;<span><panclass"GAPinput"  Aa 652]  "java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
<span class="GAPprompt">gap></span> <span class="GAPinput"># Binary code</span>
<span class="GAPprompt">gap></span> <span class=d class>56- <h5
<span class="GAPprompt">gap></span> <ss"ode">java.lang.StringIndexOutOfBoundsException: Range [54, 4) out of bounds for length 239
[ a cyclic [9,7,3]1 MDS code over GF(8), a linear [9,5,3]2..3 shortened code,
  a cyclic [9,4,6]div="<>
<span class="GAPprompt">gap></span> <span class="GAPinput">A := ExtendedCode( HammingCode(3,GF(2)) );;</span>
<"=&;/> =Pinput>/>
[linear[3,java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  a cyclic [8,7,2]1 dual code ]
<span class="GAPprompt">gap></span> <span class="java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 44
a linear [72,38,s =GAPprompt>gt<span <span classGAPinput
<>#/>
<span class="GAPprompt">gap&[ linear 8,3,4].. ,  linear[,,]  code,
<class"APprompt">&<span><span class="</>
<spanclass="GAPprompt">gap&g;/span>span = java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<></span><<class="xample><>
<span class="GAPprompt">gap></span> <span class=span ="">>panclass"GAPprompt>><span> span =GAPinput></>
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimumDistance(O1);; MinimumDistance(O2);; MinimumDistance(O3);;</span>
<span class="GAPprompt">gap>java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 140
<spancyclic973]  code(,a [9,5,2... shortenedcode
<spanclass="APprompt">gap></span> <span class="GAPinput">O  := [ O1, O2, O3 ];</span>
[ a linear [6,4,3]1 dual code, a linear [6,3,4]2..3 extended code,
  alinear 6253. extended code java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
<class>><Gjava.lang.StringIndexOutOfBoundsException: Range [24, 22) out of bounds for length 117
[ a cyclic [5,1,5]3..4 enumerated code overclassjava.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 90
   [2,2. ap;</span <span class="APinput"C=BZCodeNC(,I;</span>
  a cyclic [5,3,1..3]2a linear [72,38,2].72 BlokhZyablov concatenated
<span class="GAPprompt">gap></span> <span class="GAPinput">C  := BZCodeNC( O, I );</span>
  [0,9,5.150.  Zyablovconcatenatedcode
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimumDistance(C);</span>
15
<> <span span class="GAPprompt">gap></span> <span classjava.lang.StringIndexOutOfBoundsException: Range [109, 107) out of bounds for length 140
[]0.30 lokhconcatenated ",
  Icodes[panG"&;/pan <class"java.lang.StringIndexOutOfBoundsException: Range [63, 60) out of bounds for length 92
,2,4]2..3 enumerated code over GF(5),<java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 98
r GF(5) ]"<pan class="GAPprompt">gap></span> <span class="GAPinput">for i in [4, 5] do; MinimumDistance(Cy[i]);; od;</span>
  "Outer codes: [ a linear [6,4,3]1 dual code, a linear [6,3,4]2..3 extended c\
ode, a linear [6,2,5]3..4 extended code ]" ]
</pre></div>

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