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<title>GAP (guava) - Chapter 6: Manipulating Codes</title>
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<         ":/www..java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
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<div class="chlinkprevnexttop">span=<br/>span ">nbsp&;/>a =".tmljava.lang.StringIndexOutOfBoundsException: Range [107, 106) out of bounds for length 139

<p id="mathjaxlink" class="pcenter"><a href="chap6.span class=ContSS"</< class"ocss">&bsp;spana hrefchap6_mjhtmlX7B0A6E1F82686B43.1 </a/span
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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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< classC"<br /< =""nbsp&bsp;</pan>a href=".htmlX8799">6117 <a<panjava.lang.StringIndexOutOfBoundsException: Index 135 out of bounds for length 135
<;</span><a href="chap6_mj.html#X79577EB27BE8524B">6.< =C><br /<spanclass=nocss>&<span> =.htmlX8799"61-7 <a><span>
< ""><a href="chap6_mj.html#X7E92DC9581F96594">6.1-13 ConstructionBCode</a></span>
java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 134
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X81FE1F387DFCCB22">6.1-15 ConversionFieldCode</a></span>
<span class="ContSS"><br /><span span class="ContSS"><br /><span class"> /><class"n;b;/ chap6_mjhtmlX81FE1F387DFCCB22.-15ConversionFieldCode>span
<.#8799F4BF81B0842B117 a/>
<span class="ContSS"><br /><span class="nocss"> &span class="ContSS"><br /><span class="nocss">  java.lang.StringIndexOutOfBoundsException: Range [109, 106) out of bounds for length 137
< ""n/> hap6_mj#Ajava.lang.StringIndexOutOfBoundsException: Range [107, 106) out of bounds for length 142
<span class="ContSS"><br /><span class="<=""< >span ="nocss&bsp&bsp;/>< hrefc.X8134>.- </a</span
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< class=ContSect class=>br< class=nocss"&bsp;nbsp;/<href=chap6_mj.span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html-9 ShortenedCodea>/pan
Functions that div class="ContSect"><span class="tocline"><span class="nocss"> <Functions thatGenerateaNewCode from < class=ContSS"</ class"ocss>nbsp&;/<span class="ContSS"><br /><span class="nocss">  </spanX809376187C1525AA61- ResidueCodea<>
</java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
</span
<div""<r>spanclass"pan class="ContSS"><br /><span class="nocss;bsp;/> href=hap6_mj.X82>6-6TraceCode<java.lang.StringIndexOutOfBoundsException: Range [128, 127) out of bounds for length 135
<class""brjava.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 138
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<span class="ContSS"><br />spanclass=ContSS>br /<pan class=">;&;<span>a ="chap6_mjhtml#8228"625UnionCode/>/>
< =ContSS">brjava.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 141
<><a href="chap6_mj.html#X8228A1F57A29B8F4">625UnionCode//java.lang.StringIndexOutOfBoundsException: Index 134 out of bounds for length 134
><ajava.lang.StringIndexOutOfBoundsException: Range [74, 73) out of bounds for length 146
>span
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X7D8981AF7DFE9814">6.2-8 java.lang.StringIndexOutOfBoundsException: Index 135 out of bounds for length 63
< "C37D467791CE99B".- </a/>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X7B50943B8014134F">611BZCodea/
<span class"ontSS"</<class"ocss"< class==ontSS< <pan""&;nbsp<spanahref"html#78<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X8228A1F57A29B8F4
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</>div
</div>

<h3><> some ,it< classC> >< class=>bsp&;/<href"hap6_mjtml#>27AmalgamatedDirectSumCode/>/>

<>Inthis chapter we describe several functions <strong class=spanclass"ontSS"< /spanclass"nocss"&;java.lang.StringIndexOutOfBoundsException: Range [58, 58) out of bounds for length 0

<p>In some cases, it is faster to perform java.lang.StringIndexOutOfBoundsException: Range [0, 54) out of bounds for length 0

<p>Because class=ContSS< >span

<p>In Section

<pjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<h4java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
Functions that Generate
</spandiv class<In  < href="hap6_mj#8271A4697FDA97B2""><span java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 0

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

< java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 0

<<span>/h4>
<p><code

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

<p>A cyclic code in general is no longer

<div class="example"<span class=="GAPprompt">gap></span> <span class="GAPinput"><<code class="ode>ExtendedCode</code> extends the code <var class="a linear [7,4,]1 Hamming (3,2)codeover
<span lass"APprompt"gapgt;span span ""C1=HammingCode3 GF2))<span>
a linear [7="java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
ass"GAPprompt">ap&t/pan><span class"GAPinput">2 : ExtendedCode <span>
a linear [8,442extendedcode
<span ="">gap;<span>< class=GAPinput">IsEquivalent( C2,ReedMullerCode(1, 3 ));</pan>
true
<span class="GAPprompt">gapspan=G>><span><span =""C3: EvenWeightSubcode C1)/>
[0 ,4 ,4,,4,4,4 ,4, 4,4 4 <panclass"">java.lang.StringIndexOutOfBoundsException: Range [31, 27) out of bounds for length 112
<span class=GAPprompt>gap>/><span class"">C3 = EvenWeightSubcode C1 );</>
a linear [7,3,4]2..<pre</>
</<java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12

<p>To </pre></di

<h

<h5>6.java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
< =""<var thecolumn isdone simplycuttingoff  fromeach .  the lengthdecreases byone   distance  alsodecrease by .</>

<p>This command

<pdivpPuncturing codein resultsin anon- .If  is java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div class="example"><pre>
<"">&;/> class"Pinput"C1:BCHCode15,5 2 ;span
acyclic,7,53.  codeclass""ap<pan< ">:PuncturedCodeC </span>
< class">&;/> < class=GAPinput"C2( )<span
a linear [span=GAPprompt"gapg<> <pan =APinput"(C2 ;/java.lang.StringIndexOutOfBoundsException: Index 93 out of bounds for length 93
<span  classGspan class"GAPpromptgap&;span>span =GAPinput> C1,[12,,,,6, ;/>
false
<spanclass""gapgt</ span class=G"PuncturedCode( C1,[1paceCode( ,GF())) #The  java.lang.StringIndexOutOfBoundsException: Range [133, 132) out of bounds for length 152
a linear linear [,>/iv
<span <p><code cl"code>ExtendedCode<code> extends the  again (java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 12
alinear 3,31java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</pre>

<p>>a="87691 =""><a<p>

<=X87691""name"87691"<>/java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62

<h5>6.1-3 java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 0

<div class="func"><table java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 0
<p><code class="code">an(33,4.).

<p>Of course, if all codewords of <var class="Arg">C</var> are already of even weight, the returned code is equal to <var class=span 0,4 ,4 ,4 6 44 ,4,6 ,4,,,,8  ,6,8 4

<div class="example"><pre>
<span class="GAPprompt">java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
(,48).  weightsubcode
<span class="GAPprompt">java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 12
[0 4 ,4,4,4 4,6,4, 4,4,4,6, 4, 4java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  4, 6,4 6,8, 4, 6 8 ]
<span class="<h5>6.1-4 PermutedCodeh5>
a lineardiv =java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<<iv>

<p><code classdiv class"xample"pre

<p><a id=" linear [15,5,7]5code

<h5>6.1-4 PermutedCode linear1557]puncturedcyclic[15,5 BCH , delta7, =1over (2)

<div class="pan =<pan =G"ap;/>spanclass="GAPinput">2 : ( 15,7,GF2) ;/>
<p>code ="(2,, ,13 7110 6, ,12 5)

<div =example>pre
<span a linear<panclass"APprompt>>/> <span class""C2 =C1;/>
a linear [15,pan ="GAPprompt>gap&;</pan> span class="APinputp: < class="apgt;/pan>span class=">C3/>
<span class=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
a cyclic [15,5,7<pan class
</pan> <panclass="2C1<>
false
<span class="GAPprompt< =""gap&</<span class="GAPprompt">gap></span
(2 414 ,java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 4
<span class="><a id="X87E5849784BC60D2<p><a id="X87E5849784BC60D2" name=
a linear [15,5,7]5java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<class"unc>table " =100"<>tdclass=tdleft"codeclass"#227 /> ar =Arg"C/var varclass"rgL</ar>)/d>td ="tdright>( functionnbsp;</td<tr<>/iv
true
<><div>

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

<h5>6.1-5 ExpurgatedCode</h5>

<div class=div =ealinear54,.]11 codeexpurgated 7 sjava.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
<< code"ExpurgatedCode/ode>expurgates the  < class="></>&t bythrowingaway in var ""L/java.lang.StringIndexOutOfBoundsException: Index 151 out of bounds for length 68

<div class="example"><pre>
;<classjava.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 119
[ 1, 0, 0, 35, 105, 168, 280, 435, 435, 280span""gapgt<span>span ="", ,14,,,0 1 000 0 , java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 136
java.lang.StringIndexOutOfBoundsException: Range [11, 5) out of bounds for length 99
a linear [15,4,3..4]5..11 code, expurgated a linear [15,4,3..4]5..11 code, expurgated with"><var> after augmenting. <var class="Arg">C</var> must be a linear code, <var class="Arg">java.lang.StringIndexOutOfBoundsException: Range [0, 160) out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 94
[ 1, 0, 0, 0, 14, 0
</>div

<p>This

<p><a id="span class=>&;<span><span class=G"C1 =(6;/java.lang.StringIndexOutOfBoundsException: Range [96, 95) out of bounds for length 96

<h5span class"GAPprompt"><spanjava.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 99

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; java.lang.StringIndexOutOfBoundsException: Index 107 out of bounds for length 4
<p><code span class="GAPprompt">gap></span> <span class="span class"GAPprompt">><span span =a 8,.2] code, with 3word(java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51

<p>This command can also be called with the syntax <code class=

<div class="example"><pre>
<span class="span class="GAPprompt6.- </java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
a linear [8,4,4]2 Reed-Muller (1,3) code overp>code =<span class="GAPprom"APprompt>gap&<span><pan "GAPinput2 =( C1)spanjava.lang.StringIndexOutOfBoundsException: Index 96 out of bounds for length 95
<classGjava.lang.StringIndexOutOfBoundsException: Range [24, 22) out of bounds fo"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>Input: <var class="Arg"  [  11 ,[11 ] [111]  111 ,[1  class"g<span>< class=">=RandomLinearCode(10,,F)) (C;<>

<pOutput The code generatedby< class"SimpleMath">\(Tr_pre>div>

<p><code class="code">TraceCode</code> returns the image of

<> slow.It notseem bejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div class="example"><pre<h5>6.- CosetCode/h5>
< <span class="">C:java.lang.StringIndexOutOfBoundsException: Range [82, 81) out of bounds for length 121
a  [,4,java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
p<code class=code"CosetCodecode> returns the coset of   <var class=Arg">/var> withwithrespect word <varclass=Arg"w/var>  class=Arg>w<var> must be of the codeword type.
span class="GAPprompt">gap></span>
a linear [10,,1]1..3user definedunrestricted  over GF)
1
</pre></div>

<p>" name="8799F4BF81B0842B"</a>/>

<h5>6.1-17 CosetCode</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleftspanclass="GAPprompt>gap&t<><spanclass=GAPinput>: Codeword"011011"; c in;</pan>
<p><code class="code">CosetCode</code> returns the coset of a code <var class="Arg">C</var> with respect toa (16,,.5.. code java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 22

<>eneratinga coset  alsopossibleby simply addingthe [  ]

<ivclass=example"<re>
<span class="GAPprompt">gap></span> <span class="java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 46
a linear [7,4,3]1 Hamming (3,2) code over GF(2)
<span class="span class="GAPprompt"gap&;</pan span java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 12
false
<pan class="GAPprompt"a (12,264,6..12)3..6 code wcodewordsofweight6
a (7,16,3)1 coset code
<<h5>6e/5>
[ [ 1 1 1 ], [ 1 1 1 ], [ 1 1 1 ], [ 1 1 1 ], [ 1 1 1 ], [ 1 1 1[0,0, 0, ,0,0
  [ 1 1 1 ], [ 1 1 1 ], [ 1 1 1 <p< class=code">p><code class="code">StandardFormCode>returnsvar class=Arg"C/var>afterputting itin standardform If< class=A"<
  [ 1 1 1 ], [ 1 1 1 ], [ 1 1 1 ], [ 1 1 1 ] ]
</pre></div>

<p><a span class="GAPprompt">gap></span> <span class="GAPinput">C := ConstantWeightSubcode(N, 8);</span>

<h5>6.1-18 java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 0

<div class="func"<pan class=G"[,0,,0,,0 0030  0 0,0, 0
<p><code class="code">ConstantWeightSubcode</code> returns the subcode ofspanclass="GAPprompt"&;/>span ="GAPprompt">apgt<span> < class=GAPinput> : ExtendedTernaryGolayCode; WeightDistributioneg);</pan

<>Thisalso canbecalled withthe syntax class=GAPprompt>&</ < class="Codeword(GeneratorMat(  ))</span

<p><code class="codea 122646.12)..6 ode with codewords of weight 6

<div class="example"><pre>
< =""gap&;/pan><class=G> : NordstromRobinsonCode; WeightDistribution)<span>
[ 1, 0, 0, 0, 0, 0/pre>/iv
<span">gap&;/span>>span class="GAPinput">C := ConstantWeightSubcode(N, 8);</span>
a (16,30,6..16)5..8 code with codewords of weight 8
<span class="GAPprompt"Code</5>
[ 0, 0, 0, 0, 0, 0, 0, 0, 30, 0, 0,<ivh5.1- PiecewiseConstantCode<<h5>
<span class="GAPprompt">gap></span> <<p>f<var =Arg"></var> is a linear code,the generator matrix and parity check matrix are put in standard<>code =c"></code  acode with  <span ="SimpleMath"\n
[1,0,,0, 264 0 ,440 ,0, 24 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">C := java.lang.StringIndexOutOfBoundsException: Index 80 out of bounds for length 0
a (12,264,6..12)3..6span =GAPprompt"gap&t</>< class"">span class="GAPprompt">></span> <span class="GAPinput">"random formGF));/>
<="GAPinput"WeightDistribution)<span
[ 0, 0, 0, 0, 0, 0, 264, 0, 0, 0, 0, 0, 0 ]
</pre></div>

<p><a id="X7AA203A380BC4C79" name="X7AA203A380BC4C79"></a></p>

<h5>6.1-19 StandardFormCode</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; java.lang.Strincolor: green'>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

<1 ] [11  0 1 ], [ 1 1 1  0 ] ]
</pre></div>

<p>The first constraint is satisfied by

<p><a id="X7964BF0081CC8352" java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 0

<h4>6.2 <span class="Heading">
thatGeneratea  CodefromTwoorMore Codes
</span></h4>

<p><a id="X79E00D3A8367D65A" name="X79E00D3A8367D65A"></a></p>

<<panclass=GAPprompt>&;/pan <class="AsSSortedList()<span>

<iv =func>tableclass"unc"width"100"<>< <= C1 +C2    additiondirectsum/span>
<p><code class="code">DirectSumCode</code> returns /pre>/div>

<p>If both <var class="Arg">C1</ [<>.- </>

<p>Performing a class=GAPprompt"gap;<span>div class="func"><table class="func" width%"<tr>tdclass="tdleft"><code class="func">‣ UUVCode</code>( <var class="Arg">C1</var>, <var classjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4

<div class="example
<class""gap;/> span =G"C1 =ElementsCode( [[10,[,5],GF(7 ;<span>
<span class=GAPprompt"gap&/> <span class==GAPinput"C2:=ElementsCode( [ [,0,0] 3,3,]], GF());/spanjava.lang.StringIndexOutOfBoundsException: Index 120 out of bounds for length 120
<span class="GAPprompt"><span class="GAPprompt">gap/> span =GAPinput>C2 : RepetitionCode(,GF));/
< =G"gap&t<span>< class=GAPinput"AsSSortedList);spana cyclic[,1,]2 repetition code over GF()
[[1 0000 ,[ 1   apg;/ span classAPinputR:UUVCodeC1 );</span
<java.lang.StringIndexOutOfBoundsException: Range [11, 5) out of bounds for length 107
true
</pre></div>

<p><a id="X86E9D6DE7F1A07E6" name="X86E9D6DE7F1A07E6"a linear [8,,3]ranclass="GAPprompt">gap&t</span> <spanclass=GAPinput"R=ReedMullerCode(1,3);</span>

<h5>6.2-2 UUVCode</h5>

<div class="func"><table class="func" width="100%"><tr><td java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 0
<

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

<p>The java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 0

<div class="example"><pre>
<span java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 0
a cyclic [<iv class="xample><pre>
<;</span>< =GAPinput> =RepetitionCode4,GF2)</
a cyclic [4,1,4]2 repetition a linear[54]2.4 lexicodeover GF()
<span class="APprompt">gap></span> < a cyclic7,1.] enumeratedcode  GF(,java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
a linear [8,4,1,] 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><1 code  bygenerator matrix GF()
<span class="GAPprompt">gap></span> <span class=               \sum_{ \n B {\m d} , v+u)java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
a linear [,1,]1 code definedby generatorspan ="GAPprompt"gapg;span> span"> =WholeSpaceCode,GF(2);/>
<span class"">apgt/> span class=<pIf <span class="SimpleMath">\(L \subseteq B\)</span>, and <span class="SimpleMath">\(L\)</span> and <spanspan class""gapgtspan span
a linear [3,3,1]ppan ="gap&;/span
<java.lang.StringIndexOutOfBoundsException: Range [50, 5) out of bounds for length 99
false
<span class="GAPprompt">gap></span> <span class="GAPinput">ch5.27AmalgamatedDirectSumCode
java.lang.StringIndexOutOfBoundsException: Range [5, 6) out of bounds for length 5
<= [,151..23-oldjava.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 54
true
</pre></div>

<p< ="7java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<>62<class"class =%<>t">java.lang.StringIndexOutOfBoundsException: Range [80, 79) out of bounds for length 291

<div""< =" width"%>trtdclass""<ode="&8227;</>(< classArg></>,var =">/>,< =Arg>/ )/<classt"(nbsp;unction&bsp;)/><>/>/div>
<p>The extended span"java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 98

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

<p class="center">\[
\si L
               \<>f< ="rg>a [6,] -Muller (,)ode  GF(2java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
\]</p>

<>,hiscomputationwillnot beexecuted because itmay be too time consuming largecodes./pjava.lang.StringIndexOutOfBoundsException: Index 108 out of bounds for length 108

<pjava.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 91

<""class=GAPprompt>gt/pan< ="Pinput> := ReedMullerCode( 1, 4)</>
<span class="GAPprompt">gap></span> <span class  16586ReedMuller14   GF2<class=G">g;/ java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 71
span""&;"<>p>
<span class="GAPprompt">gap><>.-java.lang.StringIndexOutOfBoundsException: Range [34, 32) out of bounds for length 37
a cyclic [7, [22,,]   code
<span class="GAPprompt">gap>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
a
<pre>/>

<p><a id="X7E17107686A845DB" name ="<=>[

h56.-

<div class="p  java.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for length 193
<p><code class="code">AmalgamatedDirectSumCode> the     \<>

<pp java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 20

<p>If <var class="Arg">c1</var>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div class"java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 26
=GAPpromptgap&;/span> span class="APinput">c := HammingCode( 3, GF(2) );</span>
a linear [7,4,3]1 java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 0
<java.lang.StringIndexOutOfBoundsException: Range [30, 5) out of bounds for length 97
a linear [16,5,8]6 Reed-Muller (1,4) code over GF(2)
< =""gap><span <=java.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 96
23,37java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 34
<=java.lang.StringIndexOutOfBoundsException: Range [24, 22) out of bounds for length 108
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimumDistance( f );;</span>
<spanclassgt<span spanclass=java.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 91
<span class="APprompt">gap></span> <span class="GAPinput">f;</span>
a linear [22,8,3]7 java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 12
</pre></divp<a ="" name""></a></

<p><a /pre<div>

<h5>6.2-8 a id="X7C37D467791CE99B=>/>/>

<  java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 1230
<p><code class="code">>For more information on Construction toa =".html#biBSloane72">[RC72]/>./

<p>In the first case, thediv =< <java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 250

<p class="center">\[
bds = \bigcup_{1 \leq i \leq \ell} ( C_1 + (L_1)_i ) \oplus ( C_2 + (L_2)_i ),
\]p

<p>where <span class=< more information on  12729,].59  =43 =1  2java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61

<thecase<

<p class="centercyclic 12729].59 java.lang.StringIndexOutOfBoundsException: Range [36, 35) out of bounds for length 61
  b{1\ i\ () o(2_)
\]</p>

<p>The length of the new code is <span class="SimpleMath< java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 92

< java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 26
<span classjava.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 107
<( WholeSpaceCode( 6 2) )<java.lang.StringIndexOutOfBoundsException: Index 122 out of bounds for length 103
<span class="GAPprompt">gap></span> <span class="GAPinput">java.lang.StringIndexOutOfBoundsException: Index 84 out of bounds for length 73
<span class="GAPprompt">></span> <span classjava.lang.StringIndexOutOfBoundsException: Range [62, 5) out of bounds for length 89
spanjava.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 89
<pre><div

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

<h5>6.2-9 ConstructionXCode</h5>

<div class="func"><table class="func" width=true
<Considerlist< =>(\</span> codesthe <spanclass=

<p>For more information on Construction X, refer to <a href="chapBib_mj.html#biBSloane72">[SRC72]</a>.</p>

<div class="example"><pre>
<span class="GAPprompt">gapgt< "javn>
<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></spa'color: green'>3. "java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
</pre/

<<id java.lang.StringIndexOutOfBoundsException: Range [24, 23) out of bounds for length 107

<h5>6.2-10 ConstructionXXCode ><span class="GAPprompt">ga"java.lang.StringIndexOutOfBoundsException: Range [62, 60) out of bounds for length 146

<" width""<>td ="<ode">#8227 </code>( <var class=Arg><>,var =><var>  =""/,""/ar,<var=ArgA2 <<java.lang.StringIndexOutOfBoundsException: Index 274 out of bounds for length 129
<p>Consider a set of linear codes over field <span class="U: a cyclic [31,6,15]10..13 code defined by check polynomial over GF [31,615]. code defined by check polynomial over GF(2)

<p>The codewords of <span class="SimpleMath">\(C_{XX}\)</span> can<span class=GAPprompt">gap><;/pan> span class="GAPinput">1: java.lang.StringIndexOutOfBoundsException: Range [88, 87) out of bounds for length 111

<ul>
<><>span =S">\( v \ |\; 0 ;\s ">>>java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 107

</li>
mpleMath\(\ |\ \ \;0\< <class""\vin bjava.lang.StringIndexOutOfBoundsException: Range [124, 123) out of bounds for length 140

</li>
<>p<pan =">(  \ |\  \ | \ a_2  ))/span>  span class""&;<span>< "> ( * ,2 ;();()<>

</li>
<>>=java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 88

</li>
<ul>
<Formoreinformation onConstruction refer to <ahref"chapBib_mj.tmlbiBAlltop84">[All84]</a>.</p>

<div class"java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 26
<span class="APprompt>gap&t;> <pan class="GAPinput">a := PrimitiveRoot(GF(32));</span>
Z25)
<"APprompt>&t;/pan> <pan="f0=java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 36
x_1(^java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
<span class="GAPprompt">gap></span  A2acyclic [6,5,2]1 dual code" ]
x_1^5+x_1^2+Z(2)^0
<span class="GAPprompt">gap></span> <span class=6"/java.lang.StringIndexOutOfBoundsException: Range [59, 58) out of bounds for length 62
^x_14x_1^+x_1+Z(2)^
<span class="GAPprompt">gap></span17
cyclic 31,1510.13 codedefined checkpolynomialoverGF(2)
<span class="GAPprompt">gap></span> <span class="GAPinput">C3 := CheckPolCode( f0 * f5, 31, GF(2) );;
a cyclic[6,1510. java.lang.StringIndexOutOfBoundsException: Range [30, 29) out of bounds for length 68
<span class="GAPprompt">gap></span> <span class="GAPinput">C1 := UnionCode(p that  ":a cyclic [,61510.13 codedefined check polynomialover GF(),
a>    -  java.lang.StringIndexOutOfBoundsException: Range [69, 64) out of bounds for length 135
U: a cyclic [31,6,15]10..13
V: a cyclic [31java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="h5>6.2-11 BZCode</h5
a linear [10,5,4]2..4 shortened code
< =GAPprompt>gt<><class=G"= DualCode( (6,GF(2) ;/>
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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