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<body> previous fromstrong"">GAP>  be from strings from vectors whose  lie a  field  also that can added togetherand there a function <code="file"<codewhichittobviousus  in are/>

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<p>That process is called "decoding." Developing codes java.lang.StringIndexOutOfBoundsException: Range [0, 59) out of bounds for length 17

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<p>The /re>

<p>The <em>Hamming distance</em> is used extensively in coding theory. It tells us in how many positions two codewords differ. In <strong class="pkg">GUAVA</strong> the Hamming distance is implemented by a function called <code class="file">DistanceCodeword</code>.</p>

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<p>A code is fundamentally just a collection of codewords. Sometimes a code is merely a <em>set</em> of codewords. Other times a code will be the vector space generated by some small set of codewords.</p>

<p>First let's build a code that is merely a set:

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">l:=["111000", "011100", "001110", "000111", "100011", "110001", "000000", "111111" ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">m:=Codeword(l,6,GF(2));    </span>
[ [ 1 1 1 0 0 0 ], [ 0 1 1 1 0 0 ], [ 0 0 1 1 1 0 ], [ 0 0 0 1 1 1 ],
  [ 1 0 0 0 1 1 ], [ 1 1 0 0 0 1 ], [ 0 0 0 0 0 0 ], [ 1 1 1 1 1 1 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">C1:=ElementsCode(m, GF(2));</span>
a (6,8,1..6)2..3 user defined unrestricted code over GF(2)
<span class="GAPprompt">gap></span> <span class="GAPinput">IsLinearCode(C1);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">WeightDistribution(C1);</span>
[ 1, 0, 0, 6, 0, 0, 1 ]
</pre></div>

<p>In this example we first wrote out a list of strings, then converted them into codewords over GF(2). The call to <code class="file">ElementsCode</code> constructs a code from a list of elements. It is possible that the set of codewords we used actually is a vector space, but the call to <code class="file">IsLinearCode</code> says no. Finally the last function tells us that there are 6 codewords of weight 3, and one each of weights 0 and 6 in this code.</p>

<p>A very useful feature of <strong class="pkg">GUAVA</strong> is the ability to construct random codes:</p>

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">C:= RandomLinearCode(12,5,GF(2));</span>
a  [12,5,?] randomly generated code over GF(2)
</pre></div>

<p>An error-correcting code's properties are fairly well captured by three numbers which traditionally are referred to using the letters n, k and d. We ask for a random code by specifying n (the wordlength), and k (the code's dimension) as well as the field which serves as the alphabet for the code.</p>

<p>One of the most salient features of a code (a feature that determines how good it will be at correcting errors) is its minimum weight, <span class="SimpleMath">d</span>. This is the smallest weight of any nonzero word in the code. If we wish to correct <span class="SimpleMath">m</span> errors we will need to have a minimum weight of at least <span class="SimpleMath">2m+1</span>.</p>

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimumWeight(C);</span>
3
</pre></div>

<p>This particular code would be capable of correcting single bit errors.</p>

<p>Finally, one might be interested in the entire distribution of the weights of the words in a code. The weight distribution is a vector that tells us how many words there are in a code with each possible weight between <span class="SimpleMath">0</span> and <span class="SimpleMath">n</span>.</p>

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">WeightDistribution(C);</span>
[ 1, 0, 0, 2, 3, 6, 7, 6, 4, 2, 1, 0, 0 ]
</pre></div>

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