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<div class="ChapSects"><a href="chap2_mj.html#X7A489A5D79DA9E5C">2 Examples</a>
<div class="ContSect"> <a href="chap2_mj.html#X7B229D2B7E58ED2E">2.1 The 5-qubit code</a>

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<div class="ContSect"> <a href="chap2_mj.html#X87244D1D7FAA03A1">2.2 Hyperbolic codes from a file</a>

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<div class="ContSect"> <a href="chap2_mj.html#X7A1DF21D8626DA84">2.3 Randomly generated cyclic codes</a>

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<h3>2 Examples</h3>

A few simple examples illustrating the use of the package. For more information see Chapter <a href="chap4_mj.html#X7C6522597D7E72FE">4</a>

<a id="X7B229D2B7E58ED2E" name="X7B229D2B7E58ED2E"></a>

<h4>2.1 The 5-qubit code</h4>

In this example, we generate the matrix of the 5-qubit code over GF(3) with the stabilizer group generated by cyclic shifts of the operator \(X_0Z_1 \bar Z_2 \bar X_3\) which corresponds to the polynomial \(h(x)=1+x^3-x^5-x^6\) (a factor \(X_i^a\) corresponds to a monomial \(a x^{2i}\), and a factor \(Z_i^b\) to a monomial \(b x^{2i+1}\)), calculate the distance, save into a file using the function <code class="code">WriteMTXE()</code>, and read the file back in using the function <code class="code">ReadMTXE()</code>.

<div class="example"><pre>
gap> q:=3;; F:=GF(q);; 
gap> x:=Indeterminate(F,"x");; poly:=One(F)*(1+x^3-x^5-x^6);;
gap> n:=5;;
gap> mat:=QDR_DoCirc(poly,n-1,2*n,F);; #construct circulant matrix with 4 rows 
gap> Display(mat);
1 . . 1 . 2 2 . . .
. . 1 . . 1 . 2 2 .
2 . . . 1 . . 1 . 2
. 2 2 . . . 1 . . 1
gap> d:=DistRandStab(mat,100,1,0 : field:=F,maxav:=20/n);
3
gap> tmp_file_name:=Filename(DirectoryTemporary(),"n5_q3_complex.mtx");;
gap> WriteMTXE(tmp_file_name,3,mat,
> "% The 5-qubit code [[5,1,3]]_3",
> "% Generated from h(x)=1+x^3-x^5-x^6",
> "% Example from the QDistRnd GAP package" : field:=F);;
gap> lis:=ReadMTXE(tmp_file_name);; # Filename(filedir,"n5_q3_complex.mtx")
gap> lis[1]; # the field 
GF(3)
gap> lis[2]; # converted to `pair=1`
1
gap> Display(lis[3]);
1 . . 1 . 2 2 . . .
. . 1 . . 1 . 2 2 .
2 . . . 1 . . 1 . 2
. 2 2 . . . 1 . . 1
</pre></div>

The function <code class="code">WriteMTXE()</code> takes several arguments which specify the details of the output file format and the optional comments, see Section <a href="chap4_mj.html#X7E4EA2B38128F66B">4.2</a> for the details. These ensure that all information about the code is written into the file, so that for reading with the function <code class="code">ReadMTXE()</code> only the file name is needed. Output is a list: <code class="code">[field,pair,matrix,(list of comments)]</code>, where the <code class="code">pair</code> parameter describes the ordering of columns in the matrix, see <a href="chap5_mj.html#X7D0187B5831B764D">5</a>. Notice that a <code class="code">pair=2</code> or <code class="code">pair=3</code> matrix is always converted to <code class="code">pair=1</code>, i.e., with \(2n\) intercalated columns \((a_1,b_1,a_2,b_2,\ldots)\). The remaining portion is the list of comments. Notice that the 1st and the last comment lines have been added automatically.

<div class="example"><pre>
gap> lis[4];
[ "% Field: GF(3)", "% The 5-qubit code [[5,1,3]]_3",
 "% Generated from h(x)=1+x^3-x^5-x^6",
 "% Example from the QDistRnd GAP package", "% Values Z(3) are given" ]
</pre></div>

Here is the contents of the created file which illustrates the <code class="code">coordinate complex</code> data format. Here a pair \((a_{i,j},b_{i,j})\) in row \(i\) and column \(j\) is written as a row of 4 integers, "\(i\) \(j\) \(a_{i,j}\) \(b_{i,j}\)", e.g., "1 2 0 1" for the second entry in the 1st row, so that the matrix in the file has \(n\) columns, each containing a pair of integers.

<div class="example"><pre>
%%MatrixMarket matrix coordinate complex general
% Field: GF(3)
% The 5-qubit code [[5,1,3]]_3
% Generated from h(x)=1+x^3-x^5-x^6
% Example from the QDistRnd GAP package
% Values Z(3) are given
4 5 20
1 1 1 0
1 2 0 1
1 3 0 2
1 4 2 0
2 2 1 0
2 3 0 1
2 4 0 2
2 5 2 0
3 1 2 0
3 3 1 0
3 4 0 1
3 5 0 2
4 1 0 2
4 2 2 0
4 4 1 0
4 5 0 1
</pre></div>

<a id="X87244D1D7FAA03A1" name="X87244D1D7FAA03A1"></a>

<h4>2.2 Hyperbolic codes from a file</h4>

Here we read two CSS matrices from two different files which correspond to a hyperbolic code \([[80,18,5]]\) with row weight \(w=5\) and the asymptotic rate \(1/5\). Notice that <code class="code">pair=0</code> is used for both files (regular matrices).

<div class="example"><pre>
gap> filedir:=DirectoriesPackageLibrary("QDistRnd","matrices");;
gap> lisX:=ReadMTXE(Filename(filedir,"QX80.mtx"),0);;
gap> GX:=lisX[3];;
gap> lisZ:=ReadMTXE(Filename(filedir,"QZ80.mtx"),0);;
gap> GZ:=lisZ[3];;
gap> DistRandCSS(GX,GZ,100,1,2:field:=GF(2));
5
</pre></div>

Here are the matrices for a much bigger hyperbolic code \([[900,182,8]]\) from the same family. Note that the distance here scales only logarithmically with the code length (this code takes about 15 seconds on a typical notebook and will not actually be executed).

<div class="example"><pre>
gap> lisX:=ReadMTXE(Filename(filedir,"QX900.mtx"),0);;
gap> GX:=lisX[3];;
gap> lisZ:=ReadMTXE(Filename(filedir,"QZ900.mtx"),0);;
gap> GZ:=lisZ[3];;
gap> DistRandCSS(GX,GZ,1000,1,0:field:=GF(2));
8
</pre></div>

<a id="X7A1DF21D8626DA84" name="X7A1DF21D8626DA84"></a>

<h4>2.3 Randomly generated cyclic codes</h4>

As a final and hopefully somewhat useful example, the file "examples/cyclic.g" contains a piece of code searching for random one-generator cyclic codes of length \(n:=15\) over the field \(\mathop{\rm GF}(8)\), and generator weight <code class="code">wei:=6</code>. Note how the <code class="code">mindist</code> parameter and the option <code class="code">maxav</code> are used to speed up the calculation.

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