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<p id=></p>
<p><a id="X866FC1117814B64D" name="X866FC1117814B64D"></a></p>
<div class="ChapSects"><a href="chap6_mj.html#X866FC1117814B64D< class=ChapSects"<hrefchap6_mjhtmlX866FC1117814B64D6<spanclass"eading" Codesspan<a
<iv=ContSect"span class="tocline"<span ="nocss&;/>ahref=chap6_mj.tmlX8271A4697FDA97B2span=>
Functions that Generate a New Code from a Given Code
</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  <Functions thatGenerate a    a  Code
<panclass"" /n"&java.lang.StringIndexOutOfBoundsException: Range [60, 58) out of bounds for length 138
<span class="ContSS"><br /><span class="nocss"> <class""< /=>n;<java.lang.StringIndexOutOfBoundsException: Range [66, 65) out of bounds for length 138
<span class=ContSS>java.lang.StringIndexOutOfBoundsException: Range [26, 24) out of bounds for length 137
< class"ContSS"><br < class="nocss">nbsp;nbsp;span<hrefchap6_mj.#X87E5849784BC60D2">6.1-5 ExpurgatedCode</a></span>
<=ContSS>br/<class"ocss"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
< =ContSS>br /spanclass=nocss>nbsp;nbsp;/>< href=".#X7B0A6E1F82686B43">6.1-7RemovedElementsCode><>
<panclass"ContSS>br /<pan =n"> &<span<href=chap6_mj#784E1255874FCA8A"<<a id=X866FC1117814B64D "866FC1117814B64D><a/p
<l#X81CBEAFF7B9DE6EF">6.19ShortenedCode
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X7A5D5419846FC867">6.1-10 LengthenedCode</a></span>
< class=" /spanclass=""&; /span class="ContSS"><br /><span class="nocss">  </spanX784E1255874FCA8A"6.- AddedElementsCodea<span>
<pan class="">br/< class=n"nbsp; <>< =".html#X809376187C1525AA">6.1-12 ResidueCode</a></span>
<span class="ContSS"><br /><span class="nocss">  </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
<iv/
< 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
spanclass"< <span =o>&</>a ="hap6_mjhtmlX86E9D6DE7F1A07E6622UUVCode<>java.lang.StringIndexOutOfBoundsException: Index 132 out of bounds for length 132
<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
<span class="ContSS"><br /><span class="/>/iv>
</>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 for length 130
code  3word
<span class="GAPprompt">gap></span> <span class="GAPinput">C32 = ReedMullerCode
true
<class="GAPprompt>ap&;/> span =G"C1=
a linear [6,span class="gap&;/>< <span class="GAPprompt">gap></span> <span class
"GAPinput"Codeword [,,11,,])inC1;/pan>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">C2 := AugmentedCode( C1 );</span>
a linear [6,3,[10,0,,105 168,280,,435,280168 ,350 0,1 ]
<class"">&t<<span class="GAPprompt">gap></span class="APinput">C1 := HammingCode( 4 );; WeightDistribution( C1 );</span>
ue
</pre></div>

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

<p><a id="X7B0A6E1F82686B43" name="X7B0A6E1F82686B43"></a></p>

<h5>6.1-7 RemovedElementsCodespan="GAPprompt">apg;<> < class

<le class"" "%"<<classtdleft>code class"func">#;RemovedElementsCodecode ""Cvar,< =A"L/ )/<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RemovedElementsCode</code>( <var class="Arg">C</var>, <var class="Arg">L</var> )</td
<p><code class="code">RemovedElementsCode</code> returns code <var class="<span class="GAPprompt">gap></span> <span class="GAPinput">IsCyclicCode( C

<div class="example"><pre>
< class
0,,35,105 168
<span class="java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<class""ap</  =APinput> =java.lang.StringIndexOutOfBoundsException: Range [89, 87) out of bounds for length 104
a (15,2013,3..15)2..15 code with 35 word(s) removed
<="&t;/><class=">eightDistribution )</>
[ 1, 0, 0, 0, 105, 168, 280, 435, 435, 280, 168, 105, 35, 0, 0java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 0
</span> <span class"APinput>(  )     #C2  not linear,  minimum
3
</pre></div>

<p>Adding elements

<p><a id="X784E1255874FCA8A" name="X784E1255874FCA8Ajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<6.1-</java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32

<div class="func"><table class="func" width="100%"><tr><td  class="GAPprompt">gap></span> <span class="GAPinput">IsCyclicCodespanclass"APprompt"gapgt<span span > :ShortenedCode)<spanjava.lang.StringIndexOutOfBoundsException: Index 95 out of bounds for length 95
<p>a(,,.42userdefinedunrestricted   GF2)

<div class="example"><pre2
<panclass="">gap><span < ""1: (,java.lang.StringIndexOutOfBoundsException: Range [84, 83) out of bounds for length 96
acyclic [,,]n over(2panclass"gapt<span>< "Pinput>C ;/java.lang.StringIndexOutOfBoundsException: Index 89 out of bounds for length 89
<java.lang.StringIndexOutOfBoundsException: Range [0, 5) out of bounds for length 0
a(,,.63 code with1 (sadded
< =">&t</><span span class="GAPprompt">gap></span> <span class :(C5,[ 1 ,3 ;span>
true
<span class="GAPprompt">gap></span><span ="GAPprompt>&t;/>< =Gjava.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 0
a(6,,..pre>/>
<span class="GAPpromptphefunction <code class="code">LengthenedCode</code> lengthens the code again (only for linear codes), see <java.lang.StringIndexOutOfBoundsException: Index 116 out of bounds for length 0
true
</pre></div>

<p>To

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

<h5>6.1-9 ShortenedCode</h5>

<div class="func"><java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 0
A/java.lang.StringIndexOutOfBoundsException: Range [100, 99) out of bounds for length 571

<pajava.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 49

< class="GAPpromptjava.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 95

"java.lang.StringIndexOutOfBoundsException: Range [22, 5) out of bounds for length 96
<span class="alinear[,,]2. code,lengthenedwith)
1531java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 12
<span class="GAPprompt"span"APprompt"gapgt<>span=>(  ;/span>
a span class="GAPprompt">gapjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<spanjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
a431.) java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 55
< classGAPprompt>apt/span< "APinput>(  ;/>
2
<span class="GAPprompt">
(22.).2 shortened
<="">&t;<> span="AsSSortedList(C4 ;< =">>/ spanclass""C5  5 ( ;/java.lang.StringIndexOutOfBoundsException: Index 99 out of bounds for length 99
[[ 00 ,[10   ]spanclass="gap><span> spanclass""C6: ( [, ,3  ;/java.lang.StringIndexOutOfBoundsException: Index 108 out of bounds for length 108
<span class"APprompt"gapg;/
a 31a
<span class="GAPprompt">gap>/pre<divjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
linear2823,]   0, 0, 0, 0, 0, 0, 0, 0, 0 java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
<arCode( C6 );/java.lang.StringIndexOutOfBoundsException: Index 96 out of bounds for length 96
0
</pre></div>

<p>The h5>6.1-10</h5

<< "func< =unc"=100%<tr "dleft> class">8227;LengthenedCodecode>< =A"C<var[ var =></ar>])/>td ="tdright>(&bspfunctionnbsp;<td></><tablediv>

<h5>6.1-10 LengthenedCode</h5>

divclass"unc">table class"
<java.lang.StringIndexOutOfBoundsException: Range [3, 2) out of bounds for length 857

<div class="example"><pre>
< class/pre/
a linear
<p><a="" name""<a<>""gap/ panclassGAPinput>  java.lang.StringIndexOutOfBoundsException: Range [84, 82) out of bounds for length 96
a 6,]2.code  1 (
</pre></div>

ppre>iv

<java.lang.StringIndexOutOfBoundsException: Range [5, 2) out of bounds for length 62

<h5>6.1-

<class="">><a id="X7982D699803ECD0F">>p>
<p>This functionh56111s"g;/>< class=APinput> =  157)/>

<div class="example"><pre>
<=GAPpromptgap&;<span span class"APinput" =(31,)<span>
java.lang.StringIndexOutOfBoundsException: Range [24, 1) out of bounds for length 59
<s=GAPprompt>></span>span <span class="GAPprompt">gap></span> <span class: ([00010,,,,,,,,,, ,C1);/>
a linear [31,10,11]7..13 subcode
< classGAPprompt>g<>spanclass="WeightDistribution;span>
[ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 120, 190, 0, 0, 272, 255, 0, 0, 120, 66,
   0 0,0,0 ,0 ,0, 120,190 0, 0, ,272 255, 0,0,120, ,java.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 76
/>panjava.lang.StringIndexOutOfBoundsException: Range [51, 50) out of bounds for length 88
a linear [31,3,11]14..20 subcode
<span class="GAPprompt">gap></span> <span class="span ="GAPprompt"gap&;</pan><span class="GAPinput">S2:= SubCode(C, 8);</span>
[ "a linear [31,3,11]14..20 subcode of",
  "a cyclic [31,11,11]7..11 BCH code, delta
<java.lang.StringIndexOutOfBoundsException: Range [22, 5) out of bounds for length 92
[ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, "a cyclic [3111. BCH code, delta=11span =GAPprompt">ap<span>< ="C1: ( 2 )<span>
  0 0,,0,0,0,0]
</pre></div>

rjava.lang.StringIndexOutOfBoundsException: Range [7, 6) out of bounds for length 53

<h5>6.1-12 [ [ [ Operation ,   java.lang.StringIndexOutOfBoundsException: Range [51, 50) out of bounds for length 65

<div class="func"><table class="func" width="100 cyclic                    OperationC",6]]],java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
<p

<java.lang.StringIndexOutOfBoundsException: Range [5, 4) out of bounds for length 26
<span==gap;span  GAPinput>  BCHCode15  )<pan
a cyclic [upperBound=8,java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
<span class="GAPprompt">gap></span> <span<>/div
java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 30
<span class="GAPprompt<>< ="799B12F085ACB609java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 0
<span class="GAPprompt">java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
a linear [7,4,3]code class="">ConstructionBCode/java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 0
</pre</iv

<p><a id="java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26

<6.- ConstructionBCodeh5

< <>he  isalways a  ifvarclass=rgC/java.lang.StringIndexOutOfBoundsException: Range [74, 73) out of bounds for length 93
<> > a  < class"rg>C/var>is   its  code, called<em>elf-dual<em></>

<div class="example"><pre>
GC1: ( 2, ;/>
a linear [32,16,8]6 Reed-Muller (2,5) code over (n =24,k: 9 q: 2 references : rec(  )
<span =""gap&<span>       [OperationUUVCode ][[[""]
a linear [4,,]5.10 Construction (8 coordinates
<">BoundsMinimumDistance 24 9,GF()) # java.lang.StringIndexOutOfBoundsException: Range [18, 1) out of bounds for length 68
rec( n := 24, k := 9, q := 2, references := rec(  ),
  onstruction : [ [span class="GAPprompt">gap/pan><span =GAPinput> =RD;<spanjava.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 76
      [[[OperationU   :["b,9)8,uuvconstruction ofC1 and C2:,
                      [[[Operation "RepetitionCode"]  6,2]] ] ]
                  [ [ Operation "CordaroWagnerCode" ], [ 6 ] ] ] ],
          [ [ Operation "CordaroWagnerCode" ], [ 12 ] ] ] true      "b(2)4 - code,"b12,)8 Wagner"]
  java.lang.StringIndexOutOfBoundsException: Range [29, 5) out of bounds for length 75
      "Lb(12,7)=4, u u+v construction
      Lb,)2 ofthe code,
      "Lb(6,2)=4, Cordaro-Wagner code
  upperBound := 8,
  upperBoundExplanation := [ "Ub(java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
                             contradict:", "Ub>6.114DualCode</5>
<//java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12

<p><a id =""<>

<h5>6.1-14 DualCode</h5>

<p>The dual code is always linear ,even <ar="rg>C<var is non-linear./>
<p><code classjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p>The  =GAPpromptgap><span span =GAPinput>(R = ;/pan

<p>If a code <var<iv class="GAPprompt"gapgt;;/span>span class""GeneratorMat R);<>

<  8,2 Reed13 over(2
<span class="GAPprompt">gap> linear8,42dualcode
linear [,,]]Reed- (1,)code over GFjava.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
<span class="span class="GAPpromptgap&p<idX82D18907800FE3D9"name"82D18907800FE3D9">a</>
a [,442 dual
<spanclassspan class==GAPprompt"&;<span><pan =APinput"DualCode(N)= NullCode(7 ())</pan
true
<span class=GAPprompt"iv class="func><ableclass="unc width=100"><tr><td class=tdleft"<code class="func>‣TraceCode<>( <<p>Input: <var class="Arg">C</var> is a linear code defined over an extension <span class="SimpleMath">\(E\)</span> of <varvarclass=Arg>/ar  ` 'p
a cyclic [7,7,1]0 whole space codep<a id"X81FE1F387DFCCB22" name="X81FE1F387DFCCB22"></a></p>
<span class="GAPprompt">gap></span> <span class="GAPinput">java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
true
</pre></div>

<p><a id="X81FE1F387DFCCB22"

<h5>6.1-15 ConversionFieldCode</h5>

<divclass=func"<class=func" =100%>tr<td class="dleft>code class=func">#8227a  104?]randomlygenerated   GF)
<code obtained from<class"><var>java.lang.StringIndexOutOfBoundsException: Index 107 out of bounds for length 0

<p>See also <code <[0,,].  defined code  GF2java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63

<class""<>
<span =GAPprompt>><span span classjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
a cyclic
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; CosetCode</code>( <var class="p< ="<pan class=G">>/><class=GAPinput">(R  R2;>
linear[,,].  java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 55
<span class="GAPprompt">gap></span> <span class="GAPinput">Size( R ) = Size( R2 )span =GAPprompt>gap&tspan><span class"GAPinput"GeneratorMat(R2 );</span
true
<pan =G>><pre><div>
[ [ Z(2)^0, Z(2)^0, Z(2)^0, Z<p>a id"82a linear [7,4,3]1 Hamming () code over GF(2)
<span class="GAPprompt">gap>false
[ (20 0<v=f"table class="func"width=100"<>td class"dleft>code ""&8227;<code>(var class=Arg"C/ )/>td<Input:< =Arg><var  a linear defined extensionspanclass"impleMath
  [ 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2a (,1631  code
</pre></div>

<p><a id="X82D18907800FE3D9" name="X82D18907800FE3D9"></a></p>

<h5>6.1-16 TraceCode

<div class="func"><table class[[111], 11  ]   ]   11, 1,[    ]
<p>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.StringIndexOutOfBoundsException: Index 115 out of bounds for length 89
<p   10 ,     1]  111  ]java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47

<p>If <p is satisfiedby  1,the secondby codeword,the thirdby codewords 3  ,the  by  ,6 7./p>

<p>If <var class="Arg">C</var> is a java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 0

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := java.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 0
<span> <span class=GAPinput""random<span>h4java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
[2.2.2 random  java.lang.StringIndexOutOfBoundsException: Range [41, 40) out of bounds for length 51
<span class="GAPprompt">gap></span> <span class="GAPinput">Codeword( GeneratorMat( G ) );</span>
[ [0 1 0 1irectSumCode/5>
<span class="GAPprompt">gap></span>div class=, so using Leon' binary....
[[1 0 0 1 ], [010 1 ]]
</>/>

<p><a id="X7EF49A257D6DB53B"a (,71.5).5 piecewise constant

<h5>6.1-20 PiecewiseConstantCode</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code
<p><code class="code">PiecewiseConstantCode</span class="GAPprompt">gap></span> <span class =ElementsCode[[1,0,[4,] ] GF(7 ;<span>

<p>A constraint is a list of integers, and a word <span class=<pan class=Gjava.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 0

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

<div class="example">< [ 1 0 0 0 0 ], [ 1 0 3 3  ,453  3]]
ass>&t/>< ="([2  , class="&;/panclassGjava.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 107
     [ [ 0, 0 ], [ java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
the
heCcode
divclass"func"<tableclass=func width="00%"<tr>td class="java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
thecodeprogramscompiled so Leon binary.
a (5,7,1..5)1..5 piecewise constant java.lang.StringIndexOutOfBoundsException: Range [0, 40) out of bounds for length 0
<pan class=GAPprompt"></span> <span class="p<code ="code">UVCode<code returnstheso-java.lang.StringIndexOutOfBoundsException: Range [60, 58) out of bounds for length 816
[ [0   00[0 0   1 ],[01 0 00]   0 0
  [1  0 11 ] [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 [8,4,4]2 U|U+V constructionclass=GAPprompt">gap></a cyclic [7,0,7]7 enumerated code(2,
<span class="GAPprompt">gap></span> <span class="GAPinput"pre</iv>
true
</pre></

<p><a id

<h5>6.2-3 DirectProductCode</h5>

<div class="java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<dive"

<p>If <var class<pan ""gapg

<class="<>
<pan class=GAPprompt"gapgt<span>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
a (2
<span class="GAPprompt">gap></span> <span class="GAPinput">L2 := LexiCode(acyclic  7,,77enumerated  overGF(),
a linear [8,,
< class"APprompt"gapgt<span>spanclass""D=DirectProductCodeL1 ;/
ajava.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 45
</pre><div

<X78F0B1BC81FB109Cjava.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 26

<6->

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code
<p><code class="code">IntersectionCode</code> returns the intersection of codes <var class="Arg">C1</var>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

< class">java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
<span class="GAPprompt">gap&a [,,] union class=example>
[ cyclic[,7,1] enumeratedcodean><class="Pinput"c:=Codeword("010");; c in G;</span>
  a cyclic [7,6,1..2]1 enumerated code over GF(2),
  [7,3,1.4]]2.. enumerated code over GF),
  a cyclic [7,0,java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
  a cyclic [7,3,1..4]2..3 a linear [3,1,3]1 code defined by geGFrue
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
   cyclic7,,] enumeratedcodeover GF(),java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
  cyclic 74,.]  codeoverGF()]
<span class="GAPprompt">gap></span> <span class="GAPinputfalse

</pre></div>

<p>The <em>hull</em> of a linear code is thespan class=GAPprompt"gap

<p>

<java.lang.StringIndexOutOfBoundsException: Range [23, 3) out of bounds for length 24

<divclass               s{inB {m}(,  +u.
<p><code class="code">UnionCode</code> returns\/p>

<div classp>The extended directjava.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<span class="GAPprompt">gap></span> pSuppose var ="rg>L</var> is an span "impleMath>[,k_L]L)/span>code, andvarclass="rg>Bjava.lang.StringIndexOutOfBoundsException: Index 120 out of bounds for length 0
alinear [2.]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< "java.lang.StringIndexOutOfBoundsException: Range [62, 60) out of bounds for length 99
a [272943].59  code delta43,b1 over2java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
<span class="GAPprompt">gap></spanspan "GAPprompt">gap></span    java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 79
a code, java.lang.StringIndexOutOfBoundsException: Range [47, 46) out of bounds for length 65
<spanspan =""gapgt< <java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 81
a cyclic [127,15,55]41..62 BCH code, delta=55, b=1 over GF"java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 79
<span class="GAPprompt">gap></span> <span class="GAPinput">G1 := java.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 0
<span class="h5>6.2-10 ConstructionXXCode
<span class="GAPprompt">gap></span> <span class=>  "%<>< =java.lang.StringIndexOutOfBoundsException: Range [73, 72) out of bounds for length 334
a linear [127,23,1..43]35..63 code defined by generator matrix over GF(178,63..6 even weight subcode] ]
< class"APprompt"Thecodewords ofspan =S">\({X})/>can  partitioned    spanclass"impleMath>(v;|;\\\b)<> <java.lang.StringIndexOutOfBoundsException: Range [175, 169) out of bounds for length 496
43
<span
<>
  [1272247.5136.  code47 = (,
  a cyclic [127,15,55]41..62 BCH code, delta=55, b=1 java.lang.StringIndexOutOfBoundsException: Range [0, 57) out of bounds for length 0
< "java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 90
true
<span class="GAPprompt">gap><<p<>(  ;| ;a_1;\ \ a_2 ))/pan otherwisep
true
<span class="java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 5
[ a cyclic [4,1,4]2 repetition code over GF(2),
  a cyclic [17,8,6]3..6 even weight subcode ]
<span class="GAPprompt">gap></span> <span class="GAPinput">CX := ConstructionXCode(C, A);</span>
a linear
<span class="GAPprompt">gap></span> <span class="GAPinput">History(CX);</span>
[<li<p<span span class="GAPprompt">gap></span> <span class="GAPinput">f0 := MinimalPolynomial( GF(2), a^0 );</span>
  "Base codes: [ a cyclic [127,15,55]41..62 BCH code, delta=55</li>
,acyclic [12722,4751].63 BCH code delta=47,b=over (2) a linear \
[127,23,43]35..63 code defined by generator matrix over GF(2) ]",
  "Auxiliary codes: [ a cyclic [4,1,4]2 repetition code over GF(2), a cyclic [\
,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) ;/>
a cyclic<p>a id"X820327D6854A50B5" name="X820327D6854A50B5"></a></p>
<span class="
a linear [7,115.]3.23  XX code
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimumDistance(CXX);</span>
17
<panclass"APprompt">>< <java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 90
[" linear [47
  "C1p>   on Blokh-   , to < href="hapBib_mjbiBBrouwer98[ro98<<>
  "C2: a cyclic [31,6,15]10..13 code defined   "C2: a cyclic [31,6,15]10..13 code defined by
  "C3 a cyclic [31,,15]10..3 code defined by check polynomial over GF(2)",
  "A1: a linear [10,5,4]2..4 shortened code",
  A   [,,,21 dualcode ]
</pre></div>

<<>212<>

<>.-1/>

<><td class""< class="unc#227;BZCode</ode> var ="">O/,< =A"Ivar )/>td ="">&; )/>/r<table/iv
<p>Given a set of outer codes of the same

<p>Note that the set of inner codes must satisfy chain condition, i.e. <span class="SimpleMath">\(I_1 = [n, k_1, java.lang.StringIndexOutOfBoundsException: Index 115 out of bounds for length 70

<p>For more span class="GAPpromptclassGAPprompt>t/<spanclass=Pinput><pan>

<>id=" =X820327D6854A50B5>/><p>

<h5>6.2-12 BZCodeNC</h5>

<div class="func"><table class="span class="GAPprompt">gap></span"java.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 70
<p>This function is the same as <code class="code">BZCode</code>, except this version is faster as it java.lang.StringIndexOutOfBoundsException: Index 105 out of bounds for length 31

< java.lang.StringIndexOutOfBoundsException: Range [12, 10) out of bounds for length 26
<  java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 70
<span java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 86
<span class="GAPprompt">gap></span> <span class="GAPinput">#</span class="GAPprompt">gap></span> <span class="GAPinput">#</span>
<panclassG>>/><java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 153
[ a  [9,7,31 MDS over GF() a linear,]2. code,
  a cyclic<span =Gjava.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 90
<span class="GAPprompt">gap></span> <span class="GAPinput">A := ExtendedCode( span class="GAPprompt">gap></span> <span class="GAPinput">O3 := ExtendedCode(GoppaCode(java.lang.StringIndexOutOfBoundsException: Index 99 out of bounds for length 67
<span class="GAPprompt">gap<  a ljava.lang.StringIndexOutOfBoundsException: Range [12, 10) out of bounds for length 38
[ a linear [8,3,4]3..4 subcode, a linear [8,4,4<pan ="GAPprompt"gap><span"java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 117
  a cyclic [8,7,2]1 dual code ]
<gt</><"> : O,);java.lang.StringIndexOutOfBoundsException: Range [89, 88) out of bounds for length 89
  7238,].  code
<class="gap&;/span>< class=java.lang.StringIndexOutOfBoundsException: Range [62, 60) out of bounds for length 70
<span class="GAPprompt<class"java.lang.StringIndexOutOfBoundsException: Range [24, 22) out of bounds for length 99
<span class="GAPprompt">gap></span> <span class="GAPinput">#</span>
<span [ a cyclic 3.4 codeoverGF(,java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
<="GAPinput"O3: ExtendedCodeGoppaCode(53,ElementsGF5));/java.lang.StringIndexOutOfBoundsException: Index 140 out of bounds for length 140
<span class="[ "a linear [30,9,15.Blokh Zyablov  codeof",
<span class="GAPprompt">gap></span> <span"nner : pan =G"ap;/pan pan =GAPinput"C : ( ,I);/>
<pan class="GAPprompt">gap></alinear 3095.].30Blokh concatenated
<java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 117
<span span classjava.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 80
[ a linear [6,4,3]1 dual code, a linear [6,3,4][]. java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 64
java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 38
<span class=
[ r GF() <v c">&bsp;a ="hap0_mj"[op of ]</>;  < java.lang.StringIndexOutOfBoundsException: Range [108, 103) out of bounds for length 263
  a cyclic [5,24].. enumerated code over GF()
  a cyclic [5,3,1..3]2 enumerated code ode, a linear [6,2,] extended code]"]
"java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 92
java.lang.StringIndexOutOfBoundsException: Range [2, 1) out of bounds for length 58
<span class="GAPprompt">gap></span> <span class="hr /java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
15
<span class="GAPprompt">gap></span> <span class="GAPinput">History(C);</span>
[" linear [30,,150.30Blokh Zyablov concatenated  of"
  "Inner codes: [ a cyclic [5,1,5]3..4 enumerated code over GF(5), a cyclic [5\
,r5,ajava.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 7
GF5)],java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
  "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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