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<div class="ChapSects"><a href="chap4_mj.html#X7D12102B824C97B4">4 <span class="Heading">Vectors</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X788B78DE82579F3D">4.1 <span class="Heading">Creation</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X83BF7BBC84478E77">4.1-1 CVec</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X83E8D01B82498337">4.1-2 CVec</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X826347BB7B81A06D">4.1-3 CVec</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X83E046D17AB12FF0">4.1-4 CVec</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X86642DDE797387E7">4.1-5 CVec</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7CC92B9E8286B025">4.1-6 CVec</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7B9566C1867C5EA6">4.1-7 CVecClass</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X78E5FA4881DE6D1A">4.1-8 CVecClass</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X871E42B97E3664EB">4.1-9 CVecClass</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X81C506327D635947">4.1-10 CVecClass</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X79F5D11E830B9768">4.1-11 CVecClass</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X87D5ABC1835C4A31">4.1-12 CVecClass</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X876613FC7A9A276C">4.1-13 ZeroSameMutability</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7BA2BD517C058720">4.1-14 ShallowCopy</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X84EBC4427913251F">4.1-15 Randomize</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X79862FF37B75BF48">4.2 <span class="Heading">Arithmetic</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X788A3D6482290F7D">4.2-1 <span class="Heading">Handling of scalars in arithmetic operations</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7C2F565480B1A4A1"><code>4.2-2 \+</code></a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7CFEB1507E279EF2"><code>4.2-3 \-</code></a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7DD3036C847E12DC">4.2-4 AdditiveInverseSameMutability</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7A34F9AA8330EBB9">4.2-5 AdditiveInverseMutable</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X8191F74E797B4A3D"><code>4.2-6 \*</code></a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X820D829E87A82A80">4.2-7 AddRowVector</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X846D1EDA7B5B1B39">4.2-8 MultVector</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7F4B8B747E88C59B">4.2-9 ScalarProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X86115BE97842EC3B">4.2-10 ZeroMutable</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7FB1A37D7F8603B9">4.2-11 ZeroVector</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7C68FFEC84A86786">4.3 <span class="Heading">Element access and slicing</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7A6A222E87452C27">4.3-1 ELM_LIST</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X78F1C2AE7A8DD868">4.3-2 ASS_LIST</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7D12C17C873DE3B4">4.3-3 ELMS_LIST</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X78878E5E839265A6">4.3-4 CVEC_Slice</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X80EC354D78D7B5A6">4.3-5 CopySubVector</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X84347D8B7DFA790C">4.3-6 CVEC_Concatenation</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X81F3C9697E5636B6">4.4 <span class="Heading">Comparison of Vectors and other information</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X8155313286F88093">4.4-1 =</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X790D69A88340E510">4.4-2 LT</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X78952A3C788B7DB3">4.4-3 IsZero</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7F42E5AD87EC9D5A">4.4-4 PositionNonZero</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X8370979D7F451279">4.4-5 PositionLastNonZero</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X780769238600AFD1">4.4-6 Length</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X80A1A9C386FD0E1B">4.5 <span class="Heading">Changing representation, Unpacking</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7BD632E57E1EB300">4.5-1 Unpack</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X850FB74483F87453">4.5-2 IntegerRep</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7FDFEF7A7E850151">4.5-3 NumberFFVector</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7E055C467B1B54D8">4.5-4 CVecNumber</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X853A85AE878B0CBF">4.6 <span class="Heading">Access to the base field</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X854192428704857E">4.6-1 BaseDomain</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X842335E47DC49702">4.6-2 BaseField</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X81B99EEC7C134213">4.6-3 Characteristic</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X81E3BBBB80096DC2">4.6-4 DegreeFFE</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X823EFBDB7ED49554">4.7 <span class="Heading">Hashing techniques for <code class="code">cvec</code>s</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7F0228A18327F87D">4.7-1 CVEC_HashFunctionForCVecs</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X82C8DE4C78B80B1E">4.7-2 ChooseHashFunction</a></span>
</div></div>
</div>
<h3>4 <span class="Heading">Vectors</span></h3>
<p>See Section <a href="chap3_mj.html#X7ED5B368830755AF"><span class="RefLink">3.2</span></a> for a documentation of the data format of <code class="code">cvec</code>s and its restrictions.</p>
<p><a id="X788B78DE82579F3D" name="X788B78DE82579F3D"></a></p>
<h4>4.1 <span class="Heading">Creation</span></h4>
<p>Note that many functions described later in this chapter implicitly create new <code class="code">cvec</code>s, such that it is usually only in the beginning of a calculation necessary to create <code class="code">cvec</code>s explicitly.</p>
<p><a id="X83BF7BBC84478E77" name="X83BF7BBC84478E77"></a></p>
<h5>4.1-1 CVec</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CVec</code>( <var class="Arg">arg</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a new mutable <code class="code">cvec</code></p>
<p>Creates a new <code class="code">cvec</code>. See the method descriptions for details.</p>
<p><a id="X83E8D01B82498337" name="X83E8D01B82498337"></a></p>
<h5>4.1-2 CVec</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CVec</code>( <var class="Arg">cvecclass</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: a new mutable <code class="code">cvec</code></p>
<p><var class="Arg">cvecclass</var> must be a <code class="code">cvecclass</code>. Creates a new <code class="code">cvec</code> containing only zeroes. For an explanation of the term <code class="code">cvecclass</code> see Section <a href="chap3_mj.html#X7ED5B368830755AF"><span class="RefLink">3.2</span></a> and <code class="func">CVecClass</code> (<a href="chap4_mj.html#X7B9566C1867C5EA6"><span class="RefLink">4.1-7</span></a>).</p>
<p><a id="X826347BB7B81A06D" name="X826347BB7B81A06D"></a></p>
<h5>4.1-3 CVec</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CVec</code>( <var class="Arg">coeffs</var>, <var class="Arg">cvecclass</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: a new mutable <code class="code">cvec</code></p>
<p>This method takes a coefficient list and a <code class="code">cvecclass</code> as arguments. Assume the vector will be over <span class="SimpleMath">\(GF(p,d)\)</span> with <span class="SimpleMath">\(q=p^d\)</span>. For the coefficient list <var class="Arg">coeffs</var> there exist several possibilities, partially depending on the base field size. The easiest way is to use <strong class="pkg">GAP</strong> finite field elements, which will be put into the new vector in the same order. If <span class="SimpleMath">\(d=1\)</span>, one can always use <strong class="pkg">GAP</strong> immediate integers between <span class="SimpleMath">\(0\)</span> and <span class="SimpleMath">\(p-1\)</span>, the vector will then contain the corresponding cosets in <span class="SimpleMath">\(GF(p)=Z/pZ\)</span>. If <span class="SimpleMath">\(q\)</span> is small enough to be a <strong class="pkg">GAP</strong> immediate integer, then one can use <strong class="pkg">GAP</strong> immediate integers that are equal to the <span class="SimpleMath">\(p\)</span>-adic expansion using the coefficients of the representing polynomial as coefficients. That is, if an element in <span class="SimpleMath">\(GF(p,d)\)</span> is represented by the polynomial <span class="SimpleMath">\(\sum_{{i=0}}^{{d-1}} a_i x^i\)</span> with <span class="SimpleMath">\(a_i \in \{0,\ldots,p-1\}\)</span>, then one has to put <span class="SimpleMath">\(\sum_{{i=0}}^{{d-1}} a_i p^i\)</span> as integer into the coefficient list <var class="Arg">coeffs</var>. If <span class="SimpleMath">\(q\)</span> is larger, then <var class="Arg">coeffs</var> must be a list of lists of length <span class="SimpleMath">\(d\)</span> and contains for each field element of <span class="SimpleMath">\(GF(p,d)\)</span> in the vector a list of its <span class="SimpleMath">\(d\)</span> coefficients of the representing polynomial. For an explanation of the term <code class="code">cvecclass</code> see Section <a href="chap3_mj.html#X7ED5B368830755AF"><span class="RefLink">3.2</span></a> and <code class="func">CVecClass</code> (<a href="chap4_mj.html#X7B9566C1867C5EA6"><span class="RefLink">4.1-7</span></a>). Of course, the length of the list <var class="Arg">coeffs</var> must match the length of the vector given in the <code class="code">cvecclass</code>.</p>
<p><a id="X83E046D17AB12FF0" name="X83E046D17AB12FF0"></a></p>
<h5>4.1-4 CVec</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CVec</code>( <var class="Arg">coeffs</var>, <var class="Arg">p</var>, <var class="Arg">d</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: a new mutable <code class="code">cvec</code></p>
<p>This method takes a coefficient list and two positive integers <var class="Arg">p</var> and <var class="Arg">d</var> as arguments. A new <code class="code">cvec</code> over <span class="SimpleMath">\(GF(p,d)\)</span> will be created. Let <span class="SimpleMath">\(q := p^d\)</span>.</p>
<p>For a description of the possible values of the coefficient list <var class="Arg">coeffs</var> see the description of the method <code class="func">CVec</code> (<a href="chap4_mj.html#X826347BB7B81A06D"><span class="RefLink">4.1-3</span></a>).</p>
<p><a id="X86642DDE797387E7" name="X86642DDE797387E7"></a></p>
<h5>4.1-5 CVec</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CVec</code>( <var class="Arg">v</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: a new <code class="code">cvec</code></p>
<p><var class="Arg">v</var> must be a <strong class="pkg">GAP</strong> compressed vector either over <span class="SimpleMath">\(GF(2)\)</span> or <span class="SimpleMath">\(GF(p,d)\)</span> with <span class="SimpleMath">\(p^d \leq 256\)</span>. Creates a new <code class="code">cvec</code> containing the same numbers as <var class="Arg">v</var> over the field which the <code class="code">Field</code> operation returns for <var class="Arg">v</var>.</p>
<p><a id="X7CC92B9E8286B025" name="X7CC92B9E8286B025"></a></p>
<h5>4.1-6 CVec</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CVec</code>( <var class="Arg">coeffs</var>, <var class="Arg">f</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: a new mutable <code class="code">cvec</code></p>
<p>This method takes a coefficient list and a finite field <var class="Arg">f</var> as arguments. A new <code class="code">cvec</code> over <var class="Arg">f</var> will be created. Let <span class="SimpleMath">\(q := \)</span><code class="code">Size(</code><var class="Arg">f</var><code class="code">)</code>.</p>
<p>For a description of the possible values of the coefficient list <var class="Arg">coeffs</var> see the description of the method <code class="func">CVec</code> (<a href="chap4_mj.html#X826347BB7B81A06D"><span class="RefLink">4.1-3</span></a>).</p>
<p>After having encountered the concept of a <code class="code">cvecclass</code> already a few times it is time to learn how to create one. The following operation is used first to create new <code class="code">cvecclass</code>es and second to ask a <code class="code">cvec</code> for its class. In addition, it is used for <code class="code">csca</code>s.</p>
<p><a id="X7B9566C1867C5EA6" name="X7B9566C1867C5EA6"></a></p>
<h5>4.1-7 CVecClass</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CVecClass</code>( <var class="Arg">arg</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a <code class="code">cvecclass</code></p>
<p>Creates new <code class="code">cvecclass</code>es and asks <code class="code">cvec</code>s for their class. See the following method descriptions for details. For an explanation of the term <code class="code">cvecclass</code> see Section <a href="chap3_mj.html#X7ED5B368830755AF"><span class="RefLink">3.2</span></a>.</p>
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<div class="func"><table class="func" width="spanclass& a href="chap4_mjhtmlX78F1C2AE7A8DD868">4.3-2 ASS_LIST
<p>Returns: a mutable copy of <var class="Arg">v</var></p>
<ar classlass=">/var> must bea < class"code><code.>
<p<a="X84EBC4427913251F
<h5>4.-5 </h5
<div class <span class>
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