U8java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
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<ivclass""a href=".html#">6 < =""Manipulating</>/>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X8271A4697FDA97B2">d class="< ="tocline>spanclass"">&bsp<span< =chap6_mjh#">6.1 <span class=Heading">
Functions Generatea NewCodefromGiven
</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br < =ContSS><br /<span class="ocss> nbsp;</span><a href="chap6_mj.html#X7E6E4DDA79574FDB">6.1-2 PuncturedCode</a></span>
<pan =ContSS>br/<span class=nocss">&bsp; </span>< href="chap6_mj.html#X7E6E4DDA79574FDB">6.1-2 PuncturedCode</a></span>
<span class="ContSS"><brspan"">br /><spanjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="ContSS"><br /><span class="nocss"> </span><a hrefspan class"ContSS">br /span class="">java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X87E5849784BC60D2"><pan ="ContSS"</<panclass"ocss&bsp;/span>a =".htmlX784java.lang.StringIndexOutOfBoundsException: Index 106 out of bounds for length 62
java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 138
<class">/>java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 144
<>< ".html784>618</>/>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mjs class="ontSS< >spanclass"ocss"&&;</panahref"hap6_mjjava.lang.StringIndexOutOfBoundsException: Range [107, 106) out of bounds for length 137
<span class="ContSS"><br /><span class="nocss"> </span><java.lang.StringIndexOutOfBoundsException: Range [0, 68) out of bounds for length 52
<span =""< >< class=">&; />< href="chap6_mjhtml#X7982D699803ECD0F1spanclass"ontSS"> >span=nocss>nbsp <span<hrefchap6_mjhtmlXB12F085ACB609">.-4 />/>
<span class="ContSS"><br /><span class/>
<span class=span class=ContSS>br/<class=nocss>&;nbsp;/pan<href".#X794679BE7F9EB5C1">6.1-1 ExtendedCode</a></span>
<java.lang.StringIndexOutOfBoundsException: Range [38, 5) out of bounds for length 134 ""&bsp&sp</an<ahref=.#">6.-15 </a</>
<span class="ContSS"><br /><span class="span ="ontSS>br>spanclass"ocss"&;n;>ahref"span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.-17CosetCode/>/
<panclass"ontSS" /span =">nbsp;nbsp;/>ahref=chap6_mj.html#8799F4BF81B0842B>.-7CosetCode/</>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X873EA5EE85699832">6.1-18spanclass=ContSS>< >pan class="css>nbsp&bsp;span>>a=c.html#7">6.1-19 <a><span>
<span class="ContSS"span =/iv</
<spanclass"ContSS"< />span="n&<span>a =hjava.lang.StringIndexOutOfBoundsException: Range [88, 5) out of bounds for length 138
</div></div>
<java.lang.StringIndexOutOfBoundsException: Range [22, 4) out of bounds for length 145 a panclass="<br /<span =nocss">;nbsp;span>a=".#X809376187C1525AA"6.-2 /<java.lang.StringIndexOutOfBoundsException: Index 137 out of bounds for length 137
</<a
</span>
<div class="ContSSBlock"> spanclass=""> >span =n"&bsp&bsp<span<hrefspan>/
< class"span>
<span class="ontSS>< /<span =n<"> &;<span> ".#82"6.-6<>/>
<spanclass"ontSS"< /< class"">nbsp&;< =ContSS"br/><span class=nocss">;nbsp;span>< href=chap6_mjhtml#">.2-<span =ontSS"</span""n;pan>hrefc."6- UUVCode<a<>
< class"">br><pan =nocss>nbsp;nbsp;<<ahref"chap6_mj.#8228A1F57A29B8F4".- <a><span>
<span class="ContSS"><br /><span class="nocss"> </span class="ContSS"><br /><span class="nocss"> </spanspanclass"">< /><span class=< class=C"<r />span ="ocss>nbsp;&;/java.lang.StringIndexOutOfBoundsException: Range [65, 5) out of bounds for length 134
<class="< />span ="nocss">nbsp&<span>a =c.html#717107686".- <a>/>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7D8981AF7DFE9814">6.2-8 BlockwiseDirectSumCode</a></span>
<span div class"< class=tocline"< class"nocss" />a ="hap6_mj.tml#X7964BF0081CC8352">62<class<a<spanjava.lang.StringIndexOutOfBoundsException: Index 149 out of bounds for length 149
<span class="ContSS"><brspan C"br/> class=n"&bsp&bsp;/pan><ahref="ap6_mj.html#7"62-ConstructionXCode><span>
<11 <a<pan>
< class="< /><span class="nocss"> </span><a href="chap6_mj.html#X820327D6854A50B5 ="ontSS>br >span =n"<an=">br /< class=nocss>nbsp&;/span>< href=chap6_mj.X78span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X8228A1F57A29B8F4
<div</
</div>
<h3/div>/>
<p>
<>In cases is class=ontSS> > nocss">&;/pan><href=chap6_mj.#".- AmalgamatedDirectSumCode/>span
<pp>In this java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 211
<p>In java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<h4>6.1 <span class="java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
Functions> spanclass"eading"Manipulating<span>/3
</java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 0
<p><a id=p
<h5>6.1-1 ExtendedCode</h5>
<div =>Section<a href=".htmlXjava.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 0
<>java.lang.StringIndexOutOfBoundsException: Range [0, 8) out of bounds for length 0
<
<p>A cyclic code in general java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 0
<div class=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Range [24, 22) out of bounds for length 99 7431 ( java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
a linear [8,4,4]2 extended code
<class=GAPprompt>&;<span<spanclass=GAPinput>IsEquivalent , ReedMullerCode( 1, 3 ) );=">&;/><class=GAPinput> : ( , GF(2) ;/>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 47
[ 0, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4alinear8,4,]
< class"APprompt"gapgt;/ < class"APinput" =( )<span a linear [7,3,4]2. ,4 ,4 ,444 ,44 ,4,[panclass"Pprompt"gap&;</span> < class="GAPinput>IsEquivalent( C2, ReedMullerCode( 1, 3 ) );</span>
<pre></iv
<p>To undo extendingpre><div>>
<p><a id="X7E6E4DDA79574FDB linear java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>This command can also be called with the unctures var class>/>in , returnsresult.Puncturingdone offlastcolumneach.Thismeansjava.lang.StringIndexOutOfBoundsException: Range [214, 213) out of bounds for length 301
<p>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div> acyclic general in cycliccode thecodeisjava.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 0
<> <span java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 a cyclic [15,a <span cl=APprompt>gapt/ <"":BCHCodeGF);/java.lang.StringIndexOutOfBoundsException: Index 99 out of bounds for length 99 spanG>>/>spanclass"Pinput> ( )</java.lang.StringIndexOutOfBoundsException: Index 95 out of bounds for length 95
[pan =A>apt;/> spanclassGAPinput> :java.lang.StringIndexOutOfBoundsException: Range [82, 81) out of bounds for length 95
<>t<span> <class""()=C1<spanjava.lang.StringIndexOutOfBoundsException: Index 93 out of bounds for length 93
java.lang.StringIndexOutOfBoundsException: Range [32, 33) out of bounds for length 5
< "GAPprompt>&;<>span class""java.lang.StringIndexOutOfBoundsException: Range [76, 75) out of bounds for length 106 a linear
<span class =GAPprompt>&;<span>< "APinput"PuncturedCodeC1 1paceCode45 ); dimension 4to /> alinear3pre>/java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 12
<p><code class="code">ExtendedCode</code> extends the linear[,java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<< id"87691AB67FF5621B"nameX87691AB67FF5621B/>/
<56.<"87691 "A>/<>
<divp> course, all
<pspan =""gapgt;/ spanclass"GAPinput"C1: (BCHCode 8, 4, GF
<p>Of 8,34.).8evenweight
<div class="example"><pre>
rompt">&<span< =GAPinput"C1: (8,4 ()))<span
an (,33,4..).8even weightsubcode
<&;/ java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 0
[ 0,4,4 , 44,4 ,44 ,4464,,8 ,46,, 4,4 4, 6, 4, 6, 8, 4, 6, 8 ]
<span class>code class=code"></ also returns a related of weights,but withoutreducing dimension sjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 a linear [,,] Muller1,)codeoverGFa (,,.83. weight subcode
</pre></div>
< =e"<pre>
<classGAPprompt>g;java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 a [,,] a [15,5,]5 BCH code=7,b1 overGF2java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
<classAPprompt>;/> ="java.lang.StringIndexOutOfBoundsException: Range [62, 60) out of bounds for length 99 a cyclic [15,5,7]5 BCH >classc,149,,,,612 )
=APpromptgt<>span=C2 span
false
<class"gt</> < =G"> =<panGAPpromptggt/<classGAPinput>=/java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 37
< class= a15,,] permutedcode/pan < GAPinput"2=;span>
<panclass"GAPprompt>gap>/panjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</pre></div>
<p" namejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<6.1-ExpurgatedCode
<""tableunc idth100< "dleft> class=func>#227 </(< java.lang.StringIndexOutOfBoundsException: Range [135, 134) out of bounds for length 247
<p><code class="code">ExpurgatedCode</code> expurgates the code <var class="Arg">C</var>> by throwing away codewords in list <var class="Arg">L</var>. <var class="true
<class"xamplea linear[,4.4.11code,expurgated java.lang.StringIndexOutOfBoundsException: Range [48, 47) out of bounds for length 57 span="GAPprompt"gapgt;spanspan =""C1<ode=java.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for length 565
[,0,0,35,105,168,280,435435, 280 ,10535 ,0,1 java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
<span class="GAPprompt">gap></span> <pan ="GAPinput">C1 := HammingCode( 4 );; WeightDistribution( C1 );</span>
< class="APprompt"gap>/>< class="GAPinput"C2 [1 ,,014 ,0,1,0 ,00,, java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
linear15,,.45. with7 ord(java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
< =GAPprompt>&t;< span"APinput>WeightDistribution(C2 ;/span>
[ 1, 0, 0, 0, 14, 0, 0, 0, 1, 0, 0, 0, 0, java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<pre</java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 12
<p>This function does not work on non-linear codes. For removing words from a non-linear code, use <<span class="GAPprompt">gap></span> <span class="GAPinput">C2 := ExpurgatedCode( C1, L );</span>
<p><a idpThiscommand can be calledwith syntax<code =code>)/ode. When called withoutalist codewords class=code"AugmentedCode</code> returns <var class="Arg">C</var> after adding the all-ones vector to the generator matrix. <var class="Arg">C</var> must be a java.lang.StringIndexOutOfBoundsException: Index 305 out of bounds for length 0
<h5>6.[7,..2 ,augmented3word(s)
<class"gapgt;/pan/re>/div>
<p
<p>java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
<div
< ="gap>/> <span class="span"APprompt">gapg;/><panclass">([,01,111 in java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
linear8442Reed 13 over GF2java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
<java.lang.StringIndexOutOfBoundsException: Range [22, 5) out of bounds for length 130
linear[7.21 ,augmented 3word)
<span class="GAPprompt">gap
true
<span class="GAPprompt">gap></span> <span class="GAPinput">C1 := CordaroWagnerCode<The codeclass" a linear [paid"X7B0A6E1F82686B43namejava.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 0
<h561-RemovedElementsCodeh5java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
java.lang.StringIndexOutOfBoundsException: Range [48, 49) out of bounds for length 4
< "APpromptgap&<><panclass=GAPinput>2 =AugmentedCode( C1 )/> a linear
<span class="GAPprompt">divclass=examplejava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
true
</pre></div>
<pThe function<a linear [8,7,1..2]1 (java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
<div class(,20133.2. 35s java.lang.StringIndexOutOfBoundsException: Range [52, 50) out of bounds for length 101
<p><code class="GAPprompt">gap></span> <span class="GAPinput" 6,3[,,, 105168 ,435 435 ,168,105 ,01]
<div class="example"><pre>
<> <span"java.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 119
[ 1, 0, 0
<span class="java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 0
<span class="GAPprompt">gap></span> <span class="GAPinputjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 a (15,2013,3..15< ="func>table =func"=100"<>td =">code=func>‣java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
< classGAPprompt>ap&t<span><panclass
[ivclass=""<
<span< class=<div class="func"><tab" "00"<>dclass=tdleft"< =func>8227 </>( <varclass=Arg></var> var class"rg><var>)/tdjava.lang.StringIndexOutOfBoundsException: Index 185 out of bounds for length 37 3
</pre></div (,,1.)3code1 (s added
<p>java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 4
<p><a id="X784E1255874FCA8A" name="X784E1255874FCA8A"java.lang.StringIndexOutOfBoundsException: Range [0, 1) out of bounds for length 0
<h5>60,035 ,168
<div class="func"><table m acode use codeclass=""RemovedElementsCode/ (see class">&/>>< ""C2:RemovedElementsCode(C1, L ;span>
<< class""></> returns < ="rg>C< =GAPprompt"gaptpan class=>java.lang.StringIndexOutOfBoundsException: Range [81, 80) out of bounds for length 94
<div class="example"><pre>
<span class="GAPprompt">gap></span> < a cyclic[,,] overGF()
<span class="GAPprompt">gap></span> <span class="GAPinput">p""span class="GAPprompt">gap>< class=""MinimumDistanceC2) #is ,so the<span class="GAPprompt">gap></span> <span class="GAPinput">MinimumDistance( C2 ); # C2 is not linear, so the minimum =ArgC/var java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1 a6,,.6) java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 0
<span( =java.lang.StringIndexOutOfBoundsException: Range [62, 60) out of bounds for length 95
a li[4,,] shortenedcode
<span class="GAPprompt">gap></span> <panclass=G>gapgt</span> span class""C3=ElementsCode"1000,"1101,"0011"] java.lang.StringIndexOutOfBoundsException: Range [113, 111) out of bounds for length 124 a (6,4,1..span java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 0
<class"APprompt> class="APprompt&;/><panclass=GAPinput"> =NullCode 6, GF(2) );</span>
true
</pre>a cyclic [066 ullcode GF< ="gapg;/>< =""sSSortedList 4);/>
<p>To <span class>&t;/><a621.) code1word) added
<p><a id=spanclass=GAPprompt>gapg;spanspan span class="GAPprompt">gap></span> <span class= 1,2 ;java.lang.StringIndexOutOfBoundsException: Index 108 out of bounds for length 108
<h5>6.1-9 ShortenedCode</h5>
<div class (641./pre<iv
<> java.lang.StringIndexOutOfBoundsException: Range [16, 15) out of bounds for length 294
<>If varclass""C<var is non-code,< =">ShortenedCode</code> first checks which java.lang.StringIndexOutOfBoundsException: Index 117 out of bounds for length 4
<p>This command<iv class""< class""width"100%>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="example
<span class="GAPprompt">gap></span> <span class="GAPinput"><p><code class="code">ShortenedCode( C )</code> returns the code"C</r>shortened by a java.lang.StringIndexOutOfBoundsException: Range [118, 117) out of bounds for length 571 a linear [15,11,3]1 Hamming (4,2) code over GF(2)
<class=GAPprompt"gap&t;/><span ="APinput> :=ShortenedCode C1)/> a linear [14,
<span class="GAPpromptpdiv class="span class="GAPprompt">gap></span> <span class="GAPinput">C2 := LengthenedCode( C1 );</span> a (4,3,1..4)2 user defined unrestricted code overa linear [15,11,3] java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 12
< class="APprompt>><span> <span class=GAPinput"MinimumDistance C3)<span> 2
<java.lang.StringIndexOutOfBoundsException: Range [11, 5) out of bounds for length 95 a (3,2,2..3)1..2 shortened code
<span class="GAPprompt">gap></span> <span class="GAPinput =">&;java.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 91 3,3. java.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bounds for length 31
java.lang.StringIndexOutOfBoundsException: Range [13, 11) out of bounds for length 99 a linear [31,26,3]1 Hamming (5,2) code over GF(2)
< class=GAPprompt>gap&;<> < =GAPinput> =ShortenedCodeC5, [123])<span> a linear [28,23,3]2 shortened code
="APprompt>&t<java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 32
java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
<pre>/div>
<p>The function <code class=a linear [,2332 0, 0, 0, 0, 0, 0, 0, 0, 0 java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
<p><a id="java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 1
< LengthenedCode>
<class=" class=width"00"trtd class"dleft> class=func"&8227;</code> varclass"rg>/>,< class=Arg"i/ar]<td<class"tdright"&; )/></tr/></java.lang.StringIndexOutOfBoundsException: Index 249 out of bounds for length 249
<p>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="example"><pre>
<span class="GAPprompt">gap> ,0 <p<code =c"LengthenedCode C < returnscode">C<var> a linear [5,2,3]2 Cordaro-Wagner code over GF<pan</pre></div><div
<pan=G>&;span><class=GAPinput">: LengthenedCode )</java.lang.StringIndexOutOfBoundsException: Index 96 out of bounds for length 96 a linear [6,3,2]2..3code,<alinear,22.3code,lengthenedwith1column)
<>/java.lang.StringIndexOutOfBoundsException: Range [12, 13) out of bounds for length 12
<p><code class="code">ShortenedCodejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p" name=X7982D699803ECD0F"/<>
<h5>.-11=GAPpromptgap&;/pan< class""1: (, 7 )/>
<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<> ="gap&;/> <span ="APinput>=BCHCode31,)/java.lang.StringIndexOutOfBoundsException: Index 89 out of bounds for length 89
<div <span clas""gapgt;/span>< java.lang.StringIndexOutOfBoundsException: Range [51, 50) out of bounds for length 124
<span class="GAPprompt">gap></span> <span class="GAPinput">C := BCHCode(31,11);</span>
cyclic[,,].11 BCH <pan class=GAPprompt>>tspan>=APinput"WeightDistribution)<java.lang.StringIndexOutOfBoundsException: Index 92 out of bounds for length 92
< """>a<java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62 a 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
<span class="GAPprompt">gap></span> <span class="GAPinput">WeightDistribution(S1);</span>
[ 1, 0, 0,0,0,0 ,0 ,00120 ,0,0 ,255066
, ,0, 0,,0, 0 ,0,/<class"java.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 88
<class"gt/>java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 88 a linear [31,3,11]14..20 subcode
<span
[ "a linear [31,3,11<div =ejava.lang.StringIndexOutOfBoundsException: Range [22, 5) out of bounds for length 92 a[,,11]7.11 BCHcodejava.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 65
<="><><pan ="java.lang.StringIndexOutOfBoundsException: Range [81, 80) out of bounds for length 92
[,0 ,0 , ,00 ,, java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 23 0, 0, 0, 0, 0, 0, 0 ]
</pre></div>
<div class="func"><table class="<p><a id="X809376187C1525AA" name= : 24,k:= 9, :=2, references =rec(),
<p>The code class=">java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="example"><pre>
<span class[[ [java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 0 a [ ["rdaroWagnerCode , 6 ] ,
<;</spanspan class"APinput> : ResidueCode( C1;/> a linear [8,4,4]2 lowerBoundExplanation L(249=,uv constructionof and ,
<java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 0
<span class="GAPprompt">gap></span> <span class=""Lb(6,2)=4, Cordaro-Wagner code", "Lb(12,2)=8, Cordaro =GAPprompt"&t< spanclass="1: BCHCode(15,7);/pan> a linear[,4,] residuecode
<pre<div>
<<h5>.- DualCode<h5>
<code<java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 0
<div class="example"><pre>
<span class="GAPprompth5>6113</> a linear [32div<> dual is linear < A"<> -linear<p
<span class="GAPprompt"pThe<> code"< equal,is >/>/> a linear [24,9,8]5..10 Construction
<="GAPprompt"gap&;/>< class"Pinput"C1=ReedMullerCode2,5)<pan
rec n: 24 :9, =,= )
:= Ujava.lang.StringIndexOutOfBoundsException: Range [41, 40) out of bounds for length 45
java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 65
[ [a[4,,5. B8)
[ [ Operation "CordaroWagnerCode" ], [ 6 ] ] ] ],
[ [ Operation "CordaroWagnerCode" ], [ 12 ] ] ] ] =[ java.lang.StringIndexOutOfBoundsException: Range [28, 5) out of bounds for length 76
java.lang.StringIndexOutOfBoundsException: Range [24, 23) out of bounds for length 77 "Lb(12,7)=4, u u+v RepetitionCode ,[6, ,java.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 74
hejava.lang.StringIndexOutOfBoundsException: Range [41, 40) out of bounds for length 48 "(624-java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 77
upperBound>/
upperBoundExplanation
contradict:", "Ub(18,4)=8, Griesmer bound" ] )
</pre></div>
<div class="func"><p>See also <code class also/>java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
<p><code class="spanjava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 0
p dualcode is acode even if< class""/>non<p
<p>If acode <var class="Arga [8,43.java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
< s>&<<>java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 87
<span class a linear[8,4,] Reed-Muller (1,)code GF)
<class""gapgt<span <java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 89 a linear [,,]
<span class="GAPprompt">gap></ [0*(2,Za 844 Muller1, over)
true
<">gap&> = ="82>/><p a cyclic linear[4,]
<=="gap;<span =>( (,GF4 ;/pan>
true
</pre =>gapd""< f""%>trtd=><"8227 <code( <p>Input: <var class="Arg">C</var> is a linear code defined over an extension <span class="SimpleMath">\(E\)</span> of <var ="<ar is `'<>
<>a=java.lang.StringIndexOutOfBoundsException: Range [29, 27) out of bounds for length 62
<h5>6.1-15 ConversionFieldCode</h5>
<div class="func"><p>Very slow. It does not java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
<p><codediv"
<p>See also <code class=d=>table"width"%>>t"<">8227 10,] (java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
<div class="example"><pre>
<[071. java.lang.StringIndexOutOfBoundsException: Range [48, 47) out of bounds for length 63 a cyclic [4,1,4java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="GAPprompt<class""gapgt;<><=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 a linear [8,2,4]java.lang.StringIndexOutOfBoundsException: Range [24, 4) out of bounds for length 242
< java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 92
true
<span class="GAPprompt">gap></span> <span java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 0
[ (20, (a 8,4.4 , basefieldGF(java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55
<class""gap&t;</>class=> )>
[ [ Z(2)^
[ 0*Z(2), Z(2)^0, 0*Zclass=example>pre
<>/>
> id="82java.lang.StringIndexOutOfBoundsException: Range [28, 25) out of bounds for length 47
<h5>6.1-16 TraceCode</h5>
< =unc<java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 216
<java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 196
<p>Output: The linear code generated by <span class="SimpleMath">\(Tr_{E/F}(c)\)</span>, for all <span class="SimpleMath">\(c \in C\)</span>.</p>
<p><code class="code">TraceCode</code> returns the image of the code <var class="Arg">C</var> java.lang.StringIndexOutOfBoundsException: Index 98 out of bounds for length 12
<p>Very slow. It does not
[ , 111, , 11 ][1 ,
<div class="example"><pre>
<pan=gapg<<java.lang.StringIndexOutOfBoundsException: Range [52, 50) out of bounds for length 121 a [10,4,?] randomly generated code over GF(4) 5
< "java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 115 a linear [10,7,1]1..3 user defined unrestricted>:The linear span=SimpleMath\Tr_>> 1
</pre></div>
<p><a<Veryslow does seem tojava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
h56117<java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
<div class="func"><table class="func" width="100a 104,
<>codeclass=></code theofacode<class"rg><> to < "><><var""/ of java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<p>Generating a coset is also a linear [10,7,1]1..3 user defined,11.3code (2
<div class="><a id="X8799F4BF81B0842=X8799>/>/>
< java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Range [0, 1) out of bounds for length 0
< ""gapg;/spanspan ="c =("011011);c H</>
false
<class=GAPinput">C: CosetCode(,)<spanjava.lang.StringIndexOutOfBoundsException: Index 90 out of bounds for length 90 a (7,16,3)1 coset code
<span class="GAPprompt">gap></span> <span <span class="GAPprompt">gap></span> <span class>&;/pan> spanclass""WeightDistribution)/> 1]
[ 111 ], [ 111 ], [ 111 ], [ java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[ 111 ], [ 111 ], [ 111 ], [ 111 ] ]
</pre></div>
<p>This command also can be called with the syntax <code class="code">ConstantWeightSubcode
<><codeclass""</code < =><var . varclass"rg"Cjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="example"><pre>
<
[1,0 ,0 ,01120 ,0 ,0,0 ,0 ,1]
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 102 a (16,30,6..16)5..8code with codewords of weight 8
<span class="GAPprompt"div =example<> 0 ,, ,,,0 , 0 , ,0java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54
<class>&;/ span""eg=();(/>
[,0 ,0 ,0, ,0,0,440,0 ,24]
<span class="GAPprompt">gappcommand can spanclass=java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 99
(,,.123. java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 52
<span class=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[ 0, [ , 010Gjava.lang.StringIndexOutOfBoundsException: Range [24, 22) out of bounds for length 122
<>/iv>
<p><a id="span class="GAPpromptgt<><java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 102
/>
<h5>-
<p><java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
> class=C<vara , thejava.lang.StringIndexOutOfBoundsException: Range [63, 62) out of bounds for length 809
<p>If <var class="Arg">C</var> java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 0
<div class 1,0,00, 0,0,264,0,0440,0024]
<span class =java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 26
< code", ;/java.lang.StringIndexOutOfBoundsException: Range [95, 93) out of bounds for length 94 421.]. span class="GAPprompt">gap></span> <span(;
<span class="GAPprompt">gap></span> <span class="GAPinput">Codeword( GeneratorMat( G ) ); code ,using' .java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
[ [ 0101 ],
<span class="GAPprompt">gap 57,..1. java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
]
</pre></div>
<div class="func">
< java.lang.StringIndexOutOfBoundsException: Range [49, 48) out of bounds for length 153
<p>A constraint is a list of integers
<p>An example might makep< =XB "java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 0
< ">>
<span class="GAPprompt">gap&span class="GAPprompt">></span=>>>
[[0 ] , ,[,0][,2]linear21]2codejava.lang.StringIndexOutOfBoundsException: Range [46, 45) out of bounds for length 51
the C code programs are 101/>
the code programs are compiledso ' ..
the C code programs are 1 ,
the C java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 12 57,.).5piecewise
<">&;/ <java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 89
[ [ 00000 ], [ 00111 ],p
[ 11011 ], [ d=ejava.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 26
</pre></div>
< constraintjava.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<p><a id="X7964BF0081CC8352" name="X7964BF0081CC8352java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 86
<h4>6.2
Functions that Generate a New Code java.lang.StringIndexOutOfBoundsException: Range [0, 39) out of bounds for length 26
</span></h4>
< => ""="00"<> t
<>=></code java.lang.StringIndexOutOfBoundsException: Range [62, 61) out of bounds for length 463
<p>If both <var class="Arg">C1</var> and <var class="Arg">C2</var> are linear codes, the < =>gapjava.lang.StringIndexOutOfBoundsException: Range [32, 30) out of bounds for length 89
<p>Performing a direct sum can also be done by adding two codes (see Section <a href="chap4_mj.html#X832DA51986A3882C"><span class="RefLink">4. , 10110]java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 26
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 120
<cyclic 43,1
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 86
[ [ 10000 ], [ 1
;#java.lang.StringIndexOutOfBoundsException: Range [89, 87) out of bounds for length 107
true
<<div
<. java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 22
<="100>> java.lang.StringIndexOutOfBoundsException: Range [65, 64) out of bounds for length 242
<p><code class="
<p>If both <var
<p>The function <code class="code">DirectSumCode
< "java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 26
<span class="GAPprompt =>gt;<span> => := [ [0,0,[,3, ,( ;<> 4,,1
<gt;<span<class""C2=RepetitionCode4 (2)<java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0 a41,2 overGF2java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
ap;> "Pinput> =java.lang.StringIndexOutOfBoundsException: Range [75, 74) out of bounds for length 90 a linear [8,4,4]2U|U+V construction span class>><spanspan ="D= java.lang.StringIndexOutOfBoundsException: Range [74, 73) out of bounds for length 107
< ">&;<> => java.lang.StringIndexOutOfBoundsException: Range [81, 80) out of bounds for length 93
</pre></div>
<p><a id="X7BFBBA5784C293C1" name="X7BFBBA5784C293C1"></a><java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<h5>6.2-3 DirectProductCode</h5>
<div class="func"><table class="func" width>If
<>
<p>If <var class="Arg">Ci</var> is java.lang.StringIndexOutOfBoundsException: Range [0, 36) out of bounds for length 0
<e">
< G"java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 98 10,5,].42java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
<span"java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 90 a linear java.lang.StringIndexOutOfBoundsException: Range [12, 10) out of bounds for length 54
<span =java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 100
linear [1220. product
/>/
<p><a
<h5
<="funcjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<
<=ejava.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 26
<panclass=GAPprompt>&java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
[ a cyclic [7,java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 a cyclic [7,6,1..2]<=>&;/>
cyclic 731..3 linear [10,5,4]2..4 lexicode over
[,07] code (, a cyclic [7,3,1..4]2..3 enumerated code over GF 84 a cyclic [7,4,1..3]1 enumerated code over GF(2), a cyclic [7,1,/<> a p><a id="" java.lang.StringIndexOutOfBoundsException: Range [0, 33) out of bounds for length 26
<java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 0
true
</pre></div>
<p>The <em>hull</em> of a linear code is the intersection<5>6.2- IntersectionCode</h5>
<p><a ="8228A1F57A29B8F4"name"8228"></a><p
<h5>6.2-5 UnionCode</h5>
<div class="func"><a [
< of <divxample>>
<div""<pre
<span class="GAPprompt">gap&[a [71] <span class="GAPprompt">gap></spspan"java.lang.StringIndexOutOfBoundsException: Range [63, 60) out of bounds for length 99 a linear [3,2,1..2]1code defined by generator matrixspan java.lang.StringIndexOutOfBoundsException: Range [11, 3) out of bounds for length 54
<span class="span ="APprompt>ap&;<span> <=G" inU;/span>
nerator matrix over GF(java.lang.StringIndexOutOfBoundsException: Range [4, 5) out of bounds for length 4
<span class="GAPprompt">java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 0 a [7173 GF2,
<pan => 74. java.lang.StringIndexOutOfBoundsException: Range [34, 33) out of bounds for length 51
<span class="GAPprompt">gap& ""table=java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
false
<""gap
true
</pre></div>
<p><a id="h5>6.2-5 UnionCode</h5>
<h5>6.2-6 ExtendedDirectSumCode um_v } \d( )java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
<=100%>
<The java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<><classA"L/var an <class=S"\(n_L ]_\<> < ""
<p class="center">\[
\(, a [,.1 java.lang.StringIndexOutOfBoundsException: Range [36, 33) out of bounds for length 64
{\ }\d(L .
\/>
<p>Howevera [131 generator"java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 101
<> spanjava.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 307
<div a linear,,1.] - sumjava.lang.StringIndexOutOfBoundsException: Range [54, 55) out of bounds for length 54
<class"GAPprompt"gt<java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 87 out of bounds for length 47
<span class="GAPprompt">gap></span> <span class="GAPinput">java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0 a cyclic [7,7,1]0 whole space code over GF(2)
< java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 107 alinear [1,15,1..] 3-old extended direct sum code
</pre></div>
<="> =func"width=%<<class="code java.lang.StringIndexOutOfBoundsException: Range [86, 85) out of bounds for length 291
<p><code class="code">AmalgamatedDirectSumCode class=func>tableclass=func" width=100"<>< =tdleft><ode class=func>#8227 ExtendedDirectSumCode/code(<arclass=""Lvar,<varclass=Arg><var>, varclass="m<var>)/d>td class="dright>&functionn;<td/tr<table<div
<a [7,,3] Hamming (,2code overjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<> var class="">c1linear [,5](4cjava.lang.StringIndexOutOfBoundsException: Range [47, 46) out of bounds for length 52
< classp , for codes<>
<span class="GAPprompt">gap></span> <spanspanclass">gt a linear [7,4,3]1 Hamming (3,2) code over GF(2)
<">&;/> "java.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 97 a linear[,,]6 Reed- (,)code over GF()span"java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 71
<;</spanspan class=">=java.lang.StringIndexOutOfBoundsException: Range [81, 80) out of bounds for length 96 a linear [23,9,3]7/>/iv
< class=GAPprompt"gap>/id="X7D8981AF7DFE9814" name="X7D8981AF7DFE9814>
<span class=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="GAPprompt"
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 71
linear [22,,37amalgamated directsumcode
<>/>
<p>In the second bds = \bigcup_{1 \leq i \leq \ell} ( (L_1)_i \oplus (L_2)_i ).
<"\
bds = \bigcup_{1 \leq i \java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
\]</p>
<p>The lengthof <pan class"GAPprompt"g<><class"java.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 98
< =e><re
<span class="GAPprompt">gap></span> <span class=<pan class=GAPprompt>span class="GAPprompt">gap></span> <span class="GAPinput">d := ReedMullerCode( 1, 4 );</span>
<"APprompt>><span><java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 124
<span class="GAPprompt">gap></span> <span class="GAPinput">BlockwiseDirectSumCode( C1, [pan class=GAPprompt">gap></span> <span class="GAPinput">f := AmalgamatedDirectSumCode( c, d );;</span>
<span class="GAPprompt">></span> <span class="GAPinput">C2, [[ 0,0,0,0,0,0 ],[ 1,0,1,0,1,0 ]] );</span class="GAPprompt"">gap&;/span>< =GAPinput">CoveringRadius( f );; </span> a (13,1024,1..13)1..2 blockwise direct sum code
<>/java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
<p><" name"X7C37D467791CE99B"<a<p
<h5>6.2-9divnsider alist<span class="SimpleMath">\(j\)</span> linear codes of the same length <span class="SimpleMath">\(Njava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<v class=func> "unc"width"100%>tr<td class"tdleft class=f"&8227; </code( <ar"rg"<> "></> <> =""(;java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<> a <spanclass""\]/>
<>Forjava.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 106
<div class="example"><span class=""gapg;</>< class"APinput"> =BCHCode127 )<span
<span class="GAPprompt">gap></span> <span class="GAPinput">C1 := BCHCode(127,<>In second , it is defined as/p> a cyclic[127,29,4331.. BCHcode, delta=43, b=1 over GF(2)
<span class="GAPprompt">gap></span> <span class="GAPinput">C2 :=ds =\igcup_1 \eq \leq \ell} ((L_1)i \plus (_2)i ) a cyclic [127,22,47..51]36.java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 0
<pan class="span class="GAPpromptgapg<span>span class"">Append(,[()[23)<span> acyclic[1271555]41.62BCHcode,delta55,b= GF()
<span class="divclass="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Append(G1, [ GeneratorMat(C1)[23] ]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">C1 := GeneratorMatCode(G1, GF(2));</span> a linear [127,23,1..43]35..63code java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 2
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimumDistance(C1);</span> 43
<span class="GAPprompt">gap></span> <span class="GAPinput">C := [ C1, C2, C3 ];</span>
[ a linear [127,23,43]35..63code defined by generator matrix over GF(2), a cyclic [127,22,47..51]36..63 BCH code, delta=47, b=1 over GF(2), a cyclic [127,15,55]41..62 BCH code, delta=55, b=1 over GF(2) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSubset(C[1], C[2])<panclass="GAPprompt">gap></span> <span class="GAPinput<>Consider a of <span class"SimpleMath">\()/> linear of same length < java.lang.StringIndexOutOfBoundsException: Index 105 out of bounds for length 4
true
<span class="GAPpromptp a cyclic [4,1,4]2 repetition java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
true
<>/span>> span class=="Pinput>CX := ConstructionXCode(C,A)<span>
[ a cyclic [4a cyclic [27,,4331.59BCH,delta=43, =1 over GF() a cyclic [17,8,6]3..6 even weight subcode ]
<class=java.lang.StringIndexOutOfBoundsException: Range [24, 22) out of bounds for length 99
linear23]74 a cyclic [127,22,47..51]36..63 BCH code=47 =1 GF)
<class"APprompt>&;/span> span class="GAPinput23,].63 matrix (2"java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66
[ "a linear [148,23,53]43..74 Construction X code of", "Base codes: [ a cyclic [127,15,55]41..62 BCH code,>>
, a cyclic [127,22,47..51]36..63 <=java.lang.StringIndexOutOfBoundsException: Range [29, 27) out of bounds for length 62
[127,23,43]35..63code defined
: 41,]2 java.lang.StringIndexOutOfBoundsException: Range [51, 50) out of bounds for length 79
,,].6"
<>
<span=G>< codewords <spanclass"impleMathC_X\<span> be partitioned into threesections<span =""\( \;|;;|; ))</pan> where<span class="SimpleMath
<</h5
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"/li
<p>Consider a set of linear codes over field <span class="SimpleMath">\(F\)</span> of the
<ords of <span ="SimpleMath">\(C_{XX}\)</span> can be partitioned into three sections <span class="SimpleMath">\(( v\;\|\;a\;\|\;b )\)</span> <span class="GAPprompt">span class=SimpleMath"\(v\;\| \ \ \ \ ))/pan> </>
<ul>
<li<p><span class="SimpleMath">\(( v \; \| \; 0 \; \| \; 0 )\)</span> if <span class="SimpleMath">\(v \in C_4\)</span></p>
</lijava.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 81
>>java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 140
/>
<>p class cyclic127,.].BCH
<
<li><p><span class=17,,6]3.. even weight subcode]"]
</li>
</ul>
<re on XX,refer <href".#biBAlltop84"[All84]</./>
<divclass=="example">pre<gt;<span> <span class="Pinput>C3 := CheckPolCode( f0 * f5, 31, GF(2) );; MinimumDistance(C3);; Display(C3);</span>
< class="GAPprompt">gap></<div class="func"><table class="func=100%>tr< class=tdleft><ode class=="unc"‣ConstructionXXCode/"Arg"C1/var>,<var class"Arg"C2/>,<varclass="rg>C3</ar>,<var class=Arg>A1</> var class""></var>)/td><d<pan class=GAPprompt">gap></span> <span class="GAPinput">C1 := UnionCode(C2, C3);; MinimumDistance(C1);; Display(C1);</span>
Z(2^5)
<n> <span"APinput:=MinimalPolynomial GF(2) a^0 );</span>
x_1+Z(2)^0
<span class="GAPprompt">gap></span> <span
x_1^5+x_1^2+Z(2)^0
<an="Pprompt"gap;/span>span class"APinput"f5: MinimalPolynomial GF(2) a^ )/span>
x_1^5+x_1^4+x_1^2+x_1+Z(2)^0
<span classGAPprompt">\( v \;\|\; a_1 ; \ \ 0 ))/span>if span =SimpleMath>\v \in C_3\ackslash C_4\)</span></p> a cyclic [31,6,15]10..13code defined by check java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 0
<classAPpromptgap&;/<classG> =CheckPolCode *,31 ));()(;> a cyclic [31,6,15]10..13code defined by check polynomial over GF(2)
<class""gapgt;/pan <classi<>class=="\(v ;\ ;a_1 \ |\ a_2 ))/> otherwise</> a linear [31,11,11]7..11 union code of U: a cyclic [31,6,15]10..13code defined by check polynomial over GF(<> XX, =h#java.lang.StringIndexOutOfBoundsException: Range [90, 89) out of bounds for length 107
V: a cyclic [31,6,15]10..13code defined by check polynomial over GF(2)
<span class="GAPprompt">gap></span> <span class="GAPinput">A1 := BestKnownLinearCode(^) a linear [10,5,4]2..4 shortened code
<span x_1+Z()0 a cyclic [6,5,2]1 dual code
<span class="GAPprompt">gap></span> <span class="GAPinput">CXX:= ConstructionXXCode(C1, C2, C3, A1, A2 );</span>pre/> a linear [47,11,15..17]13..23 Construction XX code
<span class="GAPprompt">gapx_1^+x_1^+^x_12^ 17
<class"GAPprompt>31,]1013 java.lang.StringIndexOutOfBoundsException: Range [58, 57) out of bounds for length 68
java.lang.StringIndexOutOfBoundsException: Index 105 out of bounds for length 55 "C1: a cyclic [31,11,11]7..11 31,6,15].13code defined by check polynomial over GF(2) "C2: a cyclic [31,6,15] ":a 316,].13 by 2" "A1: a linear [10,5,4]2..4 shortened code"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 "A2: a cyclic [6,5,2]1 dual code" ]
</pre></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; spanclass"GAPprompt"gap&t;/span>span"APinput"A2 :DualCode RepetitionCode6,2))<span>
<p>Given a set of outer codes of the same length <span class="SimpleMath">\(O_i = [N, K_i, D_i
<p>Note that the set of inner codes must<pan =G">gap;<span> span class="GAPinput">History(CXX); </span>
<pspan class""gt;<>< =>":cyclic 621 java.lang.StringIndexOutOfBoundsException: Range [35, 34) out of bounds for length 37
<56.-12 BZCodeNC</h5
<iv="func"<5621 BZCode<java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 22
<="ode">div class="func"><table class="func" width="100%"><tr=tdleft>code class="">&#e='color: green'>227 </><class"Arg"<var> varclass"rg"</var> )<td< classtdright>nbspfunctionnbsp;<td>/><></>
<div class=example>pre
<span class="GAPprompt">gap></span> <span class="GAPinput">#</span>
<span class="GAPprompt">gap></span> <span class="GAPinput"># Binary code</span>
<span class="gap&t<pan>span class=APinput"</>
<span class[a [,3,]3..java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[ a cyclic [9,7,3]1 MDS codep<a id=X820327D6854A50B5name""<a></> a cyclic [9,4,6]4..5 MDS code over GFjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<panclass""gap&;/><spanclass="
<span> <span class=GAPinput#span>
[ a [34subcode a8442extended, aspan =G">ap>/>=GAPinput"#/pan
<span class="GAPprompt">gap></span> <span class="GAPinput">C ="GAPprompt&t;/span>< class=GAPinput">O2 a linear [72,38,12]0..72 span class="GAPprompt">gap <v=""<re
< classGAPpromptgap&< class="GAPprompt"gap;/ <class""#/pan>
<class="GAPprompt">gap></span> <span class="GAPinput"># Non binary code</span>
<java.lang.StringIndexOutOfBoundsException: Range [69, 5) out of bounds for length 70
<span class="GAPprompt">gap></span> <span class="GAPinput">O2 := ExtendedCode(GoppaCode(ConwayPolynomial(5,2), Elements(GFspan class="APprompt"gap&<spanspan class="GAPinput">O := [ CyclicMDSCode(2,3,7), BestKnownLinearCode(9,5,GF(2)), CyclicMDSCode(2,3,4) ];</span>
<java.lang.StringIndexOutOfBoundsException: Range [90, 5) out of bounds for length 140
<span class="GAPprompt">gap></span> <span class="GAPinput">O1 := DualCode( O3 );;</span>
<pan [,,].4extended]
<span class="GAPprompt">gap></span> <span class="GAPinput">Cy := CyclicCodes(5, GF(5));;</span>
<span class="APprompt">gap></span> <span class="GAPinput">for i in [4, 5] do; MinimumDistance(Cy[i]);; od;</span>
<span class="GAPprompt">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 extendedacyclic [,2,].3>&java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 89 a linear [6,2,5]3span class=GAPprompt>><spanspan class"GAPinput">#</span>
<pan ="APprompt">gap></span> <span class="GAPinput">I := [ Cy[5], Cy[4],linear [95.].30Blokh
[5,1,5]3.4 enumerated () a cyclic [5,2,4]2..3 enumerated span class="GAPprompt">gap></span" :ExtendedCode((ConwayPolynomial5,) (GF()));<span> a cyclic [5,3,1..3]2 enumerated code over GF(5) ]
<class"APprompt"apgt;pan> <class"GAPinput"> =BZCodeNCO, ;/spanjava.lang.StringIndexOutOfBoundsException: Index 92 out of bounds for length 92 a linear [,,.150. Zyablov code
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimumDistance(C);</span> 15
<="GAPprompt">gap></span> <span class="GAPinput">History(C);</span>
["a linear [30,9,15]0.30Blokh Zyablov concatenated code of", "Inner codes: [ a cyclic [5,1,5]3..4 enumerated code over GF(5), a cyclic [5\
,2,4]2..3 enumerated code over GF(5), a cyclic [5,3,1..3]2 5 <class=java.lang.StringIndexOutOfBoundsException: Range [31, 29) out of bounds for length 263 "Outer codes: [ a linear 5,]. 5,
linear2,3..4"
</pre></div>
<div class="chlinkprevnextbot"> <a href="chap0_mj.html">[Top of Book]</a> <a href=a linear [30,9,5..15]0..30 Blokh Zyablov concatenated code
<java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 454
<hr />
<p class="foot">generated ,2,4]2..3 enumerated code ov GF(), a java.lang.StringIndexOutOfBoundsException: Range [0, 46) out of bounds for length 7
</ (5"
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