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<div class="ChapSects"><a href="chap8_mj.html#X83FD4D318127261B">8 <span class="Heading">Applications of the Wedderga package</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8_mj.html#X8582FB957C58DFB3">8.1 <span class="Heading">Coding theory applications</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8_mj.html#X7AE55D3C7BFCF3A9">8.1-1 CodeWordByGroupRingElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8_mj.html#X7C8BBBDB78A1678E">8.1-2 CodeByLeftIdeal</a></span>
</div></div>
</div>

<h3>8 <span class="Heading">Applications of the Wedderga package</span></h3>

<p><a id="X8582FB957C58DFB3" name="X8582FB957C58DFB3"></a></p>

<h4>8.1 <span class="Heading">Coding theory applications</span></h4>

<p><a id="X7AE55D3C7BFCF3A9" name="X7AE55D3C7BFCF3A9"></a></p>

<h5>8.1-1 CodeWordByGroupRingElement</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CodeWordByGroupRingElement</code>( <var class="Arg">F</var>, <var class="Arg">S</var>, <var class="Arg">a</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: The code word of length the length of <var class="Arg">S</var> associated to the group ring element <var class="Arg">a</var>.</p>

<p>The input <var class="Arg">F</var> should be a finite field. The input <var class="Arg">S</var> is a fixed ordering of a group <span class="SimpleMath">\(G\)</span> and <var class="Arg">a</var> is an element in the group algebra <span class="SimpleMath">\(FG\)</span>.</p>

<p>Each element <span class="SimpleMath">\(c\)</span> in <span class="SimpleMath">\(FG\)</span> is of the form <span class="SimpleMath">\( c=\sum_{i=1}^n f_i g_i\)</span>, where we fix an ordering <span class="SimpleMath">\(\{g_1,g_2,...,g_n \}\)</span> of the group elements of <span class="SimpleMath">\(G\)</span> and <span class="SimpleMath">\(f_i\in F\)</span>. If we look at <span class="SimpleMath">\(c\)</span> as a codeword, we will write <span class="SimpleMath">\([f_1 f_2 ... f_n]\)</span>. (<a href="chap9_mj.html#X856D7975810BF987"><span class="RefLink">9.23</span></a>).</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">G:=DihedralGroup(8);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">F:=GF(3);;          </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">FG:=GroupRing(F,G);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">a:=AsList(FG)[27];</span>
(Z(3)^0)*<identity> of ...+(Z(3)^0)*f1+(Z(3)^0)*f2+(Z(3)^0)*f3+(Z(3)^
0)*f1*f2+(Z(3)^0)*f2*f3+(Z(3))*f1*f2*f3
<span class="GAPprompt">gap></span> <span class="GAPinput">S:=AsSet(G);</span>
[ <identity> of ..., f1, f2, f3, f1*f2, f1*f3, f2*f3, f1*f2*f3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">CodeWordByGroupRingElement(F,S,a);</span>
[ Z(3)^0, Z(3)^0, Z(3)^0, Z(3)^0, Z(3)^0, 0*Z(3), Z(3)^0, Z(3) ]

</pre></div>

<p><a id="X7C8BBBDB78A1678E" name="X7C8BBBDB78A1678E"></a></p>

<h5>8.1-2 CodeByLeftIdeal</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CodeByLeftIdeal</code>( <var class="Arg">F</var>, <var class="Arg">G</var>, <var class="Arg">S</var>, <var class="Arg">I</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: All code words of length the length of <var class="Arg">S</var> associated to the group ring elements in the ideal <var class="Arg">I</var> of <var class="Arg">FG</var>.</p>

<p>The input <var class="Arg">F</var> should be a finite field. The input <var class="Arg">S</var> is a fixed ordering of a group <span class="SimpleMath">\(G\)</span> and <var class="Arg">I</var> is a left ideal of the group algebra <span class="SimpleMath">\(FG\)</span>.</p>

<p>Each element <span class="SimpleMath">\(c\)</span> in <span class="SimpleMath">\(FG\)</span> is of the form <span class="SimpleMath">\( c=\sum_{i=1}^n f_i g_i\)</span>, where we fix an ordering <span class="SimpleMath">\(\{g_1,g_2,...,g_n \}\)</span> of the group elements of <span class="SimpleMath">\(G\)</span> and <span class="SimpleMath">\(f_i\in F\)</span>. If we look at <span class="SimpleMath">\(c\)</span> as a codeword, we will write <span class="SimpleMath">\([f_1 f_2 ... f_n]\)</span>. (<a href="chap9_mj.html#X856D7975810BF987"><span class="RefLink">9.23</span></a>).</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">G:=DihedralGroup(8);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">F:=GF(3);;          </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">FG:=GroupRing(F,G);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">S:=AsSet(G);</span>
[ <identity> of ..., f1, f2, f3, f1*f2, f1*f3, f2*f3, f1*f2*f3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">H:=StrongShodaPairs(G)[5][1];</span>
Group([ f1*f2*f3, f3 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">K:=StrongShodaPairs(G)[5][2];</span>
Group([ f1*f2 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">N:=Normalizer(G,K);</span>
Group([ f1*f2*f3, f3 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">epi:=NaturalHomomorphismByNormalSubgroup(N,K);</span>
[ f1*f2*f3, f3 ] -> [ f1, f1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">QHK:=Image(epi,H);</span>
Group([ f1, f1 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">gq:=MinimalGeneratingSet(QHK)[1];</span>
f1
<span class="GAPprompt">gap></span> <span class="GAPinput">C:=CyclotomicClasses(Size(F),Index(H,K))[2];</span>
[ 1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">e:=PrimitiveIdempotentsNilpotent(FG,H,K,C,[epi,gq]);   </span>
[ (Z(3)^0)*<identity> of ...+(Z(3))*f3+(Z(3)^0)*f1*f2+(Z(3))*f1*f2*f3, 
  (Z(3)^0)*<identity> of ...+(Z(3))*f3+(Z(3))*f1*f2+(Z(3)^0)*f1*f2*f3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">FGe := LeftIdealByGenerators(FG,[e[1]]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">V := VectorSpace(F,CodeByLeftIdeal(F,G,S,FGe));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">B := Basis(V);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage("guava");;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">code := GeneratorMatCode(B,F);</span>
a linear [8,2,1..4]4..5 code defined by generator matrix over GF(3)
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimumDistance(code);</span>
4

</pre></div>


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