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title: '一文搞懂奇偶校验、海明码和CRC',
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pageFillDescription: '1. 为什么要有校验码？, 2. 奇偶校验, 校验位的计算, 举个栗子🌰, 总结, 2. CRC（循环冗余校验）, 校验位的计算, 举个栗子🌰, 3. 海明校验码, 校验位与数据位的关系, 举个栗子🌰, 栗子2🌰, 最终总结为什么要有校验码因为在数据存取和传送的过程中由于元器件或者噪音的干扰等原因会出现错误这个时候我们就需要采取相应的措施发现并纠正错误对于错误的检测和校正大多采取冗余校验的思想即除原数据外额外增加若干位编码这些新增的代码称为校验位奇偶校验是一种常用的错误检测方法用于校验代码传输的正确性它根据被传输的一组二进制代码的数位中的个数是奇数还是偶数来进行校验如果采用奇数的方式则称为奇校验反之则称为偶校验校验位的计算奇校验如果数据单元中的数量已经是奇数则校验位设置为否则校验位设置为偶校验如果数据单元中的数量已经是偶数则校验位设置为否则校验位设置为举个栗子一个七位数的数据单元奇校验编码偶校验编码红色是校验位可以在数据的前面也可以在数据的后面总结奇偶校验是一种简单且易于实现的错误检测技术虽然它不能解决所有的错误情况但其低成本和实用性使它成为许多通信和存储系统的首选错误检测方法随着技术的发展更复杂的错误检测和纠正算法被开发出来但奇偶校验仍然是计算机科学教育和初级通信系统中的重要组成部分循环冗余校验循环冗余校验是一种根据网络数据包或计算机文件等数据产生简短固定位数校验码的一种信道编码技术主要用来检测或校验数据传输或者保存后可能出现的错误它是利用除法及余数的原理来作错误侦测的校验位的计算首先通过给定的生成多项式来确定除数然后将要进行校验或者编码的数据进行模除法异或取最后的余数然后与数据为拼接即可举个栗子假设要传输的数据为给定的生成多项式为根据题目中的生成多项式可以得出除数为下面开始进行计算计算是首先要进行补零零的个数取决于生成多项式的最高次幂经过上面的计算得出校验码位所以实际发送的数据为当客户端收到这个数据之后要进行校验经过计算得出的余数为所以说明这个数据没有差错这就是整个编码以及检错的过程海明校验码又称汉明码是一种在计算机通讯中用于错误检测和纠正的编码方法它是由理查德海明于年发明的它的核心原理是将数据分割成固定长度的块每个块都用二进制表示并在每个块中添加额外的校验位这些校验位能够检测和纠正一个或多个位的错误校验位与数据位的关系代表数据位代表校验位校验位所在的位置是其余都是数据位举个栗子假设原始数据为位置原始信息位根据上述公式当信息位时校验位最小可以取所以可以得到如下表格位置新的信息位下面开始计算校验位的值十进制对应二进制的数字如下位置二进制码海明码出现的位置对应位置上的值异或结果这里计算的规律是拿举例在第一位对应的二进制编码就是所以我们要根据上面的进制表找到第一位也是的对应的位置有然后在找到这几个位置对应的数据位求异或即可得到对应的海明码其它以此类推通过上面的计算我们就可以得到一个经过编码之后的数据位置最终信息位接收端收到对应的数据之后开始校验数据是否有错误假设传输为继续根据上面的方式重新计算校验位计算的结果为然后拿之前的校验码和新算出来的校验码求异或所以可以判断出是第位出现了错误修改之后就完成了纠错以上就是海明码检错以及纠错的过程如果你认真的看到这里你一定会有一个疑问如果出现两个以上的错误时还能纠错吗下面实践一下栗子假设出现了两个数据位的错误传输为位置最终信息位再次计算校验位为再次与之前的校验码求异或按照这次的计算结果来看是第三位数据出现了问题但是很明显我们出错的是第位但是当你把第三位修改之后再次进行校验发现没有问题了所以通过上面的第二个例子我们可以直到海明码只能检测一位错误当出现两个以上的错误是纠错结果会变得不准确最终总结特点奇偶校验海明码原理在数据中添加一个额外的校验位使整个数据的的个数为奇数奇校验或偶数偶校验用于检测单个比特的错误利用除法操作在发送的数据后面附加一个固定长度的校验码码接收端通过相同的除法操作验证数据的完整性通过在数据中添加冗余位校验位使得整个码字中特定位置的位数的个数具有确定的奇偶性以便检测和纠正错误错误检测能力只能检测单个比特的错误对于两个或更多比特的错误可能无法检测能够检测并纠正多个比特的错误具体取决于码的长度可以检测并纠正单个比特的错误有时还能检测并纠正两个比特的错误错误纠正能力无错误纠正能力根据码的长度和算法设计可以纠正一定数量的错误可以纠正单个比特的错误开销仅需添加一个校验位开销最小需要添加固定长度的码开销适中需要添加多个校验位开销相对较大实现复杂度最简单只需进行简单的位运算中等复杂度需要选择合适的多项式和实现除法操作相对较高需要设计合适的校验位位置和计算校验位的值应用场景常用于简单的数据传输和存储系统对错误检测和纠正要求不高的场合广泛应用于网络通信数据存储等领域用于确保数据的完整性适用于对错误检测和纠正有较高要求的应用如内存错误检测笔记到这里就结束啦你学会了么',
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datetime="2024-03-31T12:26:34.000Z" title="更新于 2024-03-31 20:26:34">2024-03-31</time></header><h1>1. 为什么要有校验码？</h1>
<div class="note default no-icon flat"><p> 因为在数据存取和传送的过程中，由于元器件或者噪音的干扰等原因会出现错误，这个时候我们就需要采取相应的措施，发现并纠正错误，对于错误的检测和校正，大多采取“冗余校验”的思想，即除原数据外，额外增加若干位编码，这些新增的代码称为校验位。</p>
</div>
<h1>2. 奇偶校验</h1>
<div class="note info no-icon flat"><p> 是一种常用的错误检测方法，用于校验代码传输的正确性。它根据被传输的一组二进制代码的数位中“1”的个数是奇数还是偶数来进行校验。如果采用奇数的方式，则称为奇校验；反之，则称为偶校验。</p>
</div>
<h2 id="校验位的计算">校验位的计算</h2>
<ul>
<li>
<mark class="hl-label orange">奇校验</mark> ：如果数据单元中1的数量已经是奇数，则校验位设置为0；否则，校验位设置为1。
</li>
<li>
<mark class="hl-label pink">偶校验</mark> ：如果数据单元中1的数量已经是偶数，则校验位设置为0；否则，校验位设置为1。
</li>
</ul>
<h2 id="举个栗子🌰">举个栗子🌰</h2>
<p>一个七位数的数据单元1011011</p>
<ul>
<li>奇校验编码：<span class='p red'>0</span>1011011</li>
<li>偶校验编码：<span class='p red'>1</span>1011011</li>
</ul>
<div class="note warning no-icon flat"><p>红色是校验位，可以在数据的前面也可以在数据的后面</p>
</div>
<h2 id="总结">总结</h2>
<div class="note info no-icon flat"><p> 奇偶校验是一种简单且易于实现的错误检测技术。虽然它不能解决所有的错误情况，但其低成本和实用性使它成为许多通信和存储系统的首选错误检测方法。随着技术的发展，更复杂的错误检测和纠正算法被开发出来，但奇偶校验仍然是计算机科学教育和初级通信系统中的重要组成部分。</p>
</div>
<h1>2. CRC（循环冗余校验）</h1>
<div class="note default no-icon flat"><p> 循环冗余校验（Cyclic Redundancy Check， CRC）是一种根据网络数据包或计算机文件等数据产生简短固定位数校验码的一种信道编码技术，主要用来检测或校验数据传输或者保存后可能出现的错误。它是利用除法及余数的原理来作错误侦测的。</p>
</div>
<h2 id="校验位的计算-v2">校验位的计算</h2>
<ul>
<li>首先通过给定的生成多项式来确定除数，然后将要进行校验或者编码的数据进行模2除法（异或），取最后的余数然后与数据为拼接即可</li>
</ul>
<h2 id="举个栗子🌰-v2">举个栗子🌰</h2>
<p>假设要传输的数据为1101011011，给定的生成多项式为G(X)=x<sup>4</sup>+x+1</p>
<p>根据题目中的生成多项式可以得出除数为10011，下面开始进行计算：</p>
<div class="note danger no-icon flat"><p>计算是首先要进行补零，零的个数取决于生成多项式的最高次幂</p>
</div>
<figure class="highlight plaintext"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br></pre></td><td class="code"><pre><span class="line"> 1100001010</span><br><span class="line"> _______________</span><br><span class="line">10011 ) 11010110110000</span><br><span class="line"> 10011.........</span><br><span class="line"> 1001110110000</span><br><span class="line"> 10011........</span><br><span class="line"> 10110000</span><br><span class="line"> 10011...</span><br><span class="line"> 101000</span><br><span class="line"> 10011.</span><br><span class="line"> 1110</span><br></pre></td></tr></table></figure>
<p>经过上面的计算得出校验码位1110，所以实际发送的数据为11010110111110</p>
<p>当客户端收到这个数据之后，要进行CRC校验</p>
<figure class="highlight plaintext"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br></pre></td><td class="code"><pre><span class="line"> 110000101</span><br><span class="line"> _________________</span><br><span class="line">10011 ) 11010110111110</span><br><span class="line"> 10011.........</span><br><span class="line"> 1001110111110</span><br><span class="line"> 10011........</span><br><span class="line"> 10111110</span><br><span class="line"> 10011...</span><br><span class="line"> 100110</span><br><span class="line"> 10011.</span><br><span class="line"> 0</span><br><span class="line"></span><br></pre></td></tr></table></figure>
<div class="note success no-icon flat"><p>经过计算，得出的余数为0，所以说明这个数据没有差错，这就是CRC整个编码以及检错的过程</p>
</div>
<h1>3. 海明校验码</h1>
<div class="note default no-icon flat"><p> 又称汉明码，是一种在计算机通讯中用于错误检测和纠正的编码方法。它是由理查德·海明于1950年发明的。它的核心原理是将数据分割成固定长度的块，每个块都用二进制表示，并在每个块中添加额外的校验位。这些校验位能够检测和纠正一个或多个位的错误。</p>
</div>
<h2 id="校验位与数据位的关系">校验位与数据位的关系</h2>
<ul>
<li>2<sup>k</sup>−1>=m+k</li>
</ul>
<div class="note info no-icon flat"><p>m代表数据位，k代表校验位，校验位所在的位置是2<sup>n</sup>，其余都是数据位</p>
</div>
<h2 id="举个栗子🌰-v3">举个栗子🌰</h2>
<p>假设原始数据为01101110</p>
<table>
<thead>
<tr>
<th style="text-align:center">位置</th>
<th style="text-align:center">8</th>
<th style="text-align:center">7</th>
<th style="text-align:center">6</th>
<th style="text-align:center">5</th>
<th style="text-align:center">4</th>
<th style="text-align:center">3</th>
<th style="text-align:center">2</th>
<th style="text-align:center">1</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:center">原始信息位</td>
<td style="text-align:center">0</td>
<td style="text-align:center">1</td>
<td style="text-align:center">1</td>
<td style="text-align:center">0</td>
<td style="text-align:center">1</td>
<td style="text-align:center">1</td>
<td style="text-align:center">1</td>
<td style="text-align:center">0</td>
</tr>
</tbody>
</table>
<p>根据上述公式，当 m（信息位）=8时，k（校验位）最小可以取 4 ，所以可以得到如下表格：</p>
<table>
<thead>
<tr>
<th style="text-align:center">位置</th>
<th style="text-align:center">12</th>
<th style="text-align:center">11</th>
<th style="text-align:center">10</th>
<th style="text-align:center">9</th>
<th style="text-align:center">8</th>
<th style="text-align:center">7</th>
<th style="text-align:center">6</th>
<th style="text-align:center">5</th>
<th style="text-align:center">4</th>
<th style="text-align:center">3</th>
<th style="text-align:center">2</th>
<th style="text-align:center">1</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:center">新的信息位</td>
<td style="text-align:center">0</td>
<td style="text-align:center">1</td>
<td style="text-align:center">1</td>
<td style="text-align:center">0</td>
<td style="text-align:center">H8</td>
<td style="text-align:center">1</td>
<td style="text-align:center">1</td>
<td style="text-align:center">1</td>
<td style="text-align:center">H4</td>
<td style="text-align:center">0</td>
<td style="text-align:center">H2</td>
<td style="text-align:center">H1</td>
</tr>
</tbody>
</table>
<p>下面开始计算校验位的值</p>
<p>十进制对应二进制的数字如下</p>
<table>
<thead>
<tr>
<th style="text-align:center">位置</th>
<th style="text-align:center">二进制码</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:center">1</td>
<td style="text-align:center">0001</td>
</tr>
<tr>
<td style="text-align:center">2</td>
<td style="text-align:center">0010</td>
</tr>
<tr>
<td style="text-align:center">3</td>
<td style="text-align:center">0011</td>
</tr>
<tr>
<td style="text-align:center">4</td>
<td style="text-align:center">0100</td>
</tr>
<tr>
<td style="text-align:center">5</td>
<td style="text-align:center">0101</td>
</tr>
<tr>
<td style="text-align:center">6</td>
<td style="text-align:center">0110</td>
</tr>
<tr>
<td style="text-align:center">7</td>
<td style="text-align:center">0111</td>
</tr>
<tr>
<td style="text-align:center">8</td>
<td style="text-align:center">1000</td>
</tr>
<tr>
<td style="text-align:center">9</td>
<td style="text-align:center">1001</td>
</tr>
<tr>
<td style="text-align:center">10</td>
<td style="text-align:center">1010</td>
</tr>
<tr>
<td style="text-align:center">11</td>
<td style="text-align:center">1011</td>
</tr>
<tr>
<td style="text-align:center">12</td>
<td style="text-align:center">1100</td>
</tr>
</tbody>
</table>
<table>
<thead>
<tr>
<th style="text-align:center">海明码</th>
<th style="text-align:center">出现的位置</th>
<th style="text-align:center">对应位置上的值异或</th>
<th style="text-align:center">结果</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:center">H1</td>
<td style="text-align:center">3，5，7，9，11</td>
<td style="text-align:center">0⊕1⊕1⊕0⊕1</td>
<td style="text-align:center">1</td>
</tr>
<tr>
<td style="text-align:center">H2</td>
<td style="text-align:center">3，6，7，10，11</td>
<td style="text-align:center">0⊕1⊕1⊕1⊕1</td>
<td style="text-align:center">0</td>
</tr>
<tr>
<td style="text-align:center">H4</td>
<td style="text-align:center">5，6，7，12</td>
<td style="text-align:center">1⊕1⊕1⊕0</td>
<td style="text-align:center">1</td>
</tr>
<tr>
<td style="text-align:center">H8</td>
<td style="text-align:center">9，10，11，12</td>
<td style="text-align:center">0⊕1⊕1⊕0</td>
<td style="text-align:center">0</td>
</tr>
</tbody>
</table>
<div class="note primary no-icon flat"><p>这里计算的规律是，拿H1举例，H1在第一位，对应的二进制编码就是 1，所以我们要根据上面的进制表，找到第一位也是 1 的对应的位置，有 3,5,7,9,11，然后在找到这几个位置对应的数据位求异或，即可得到H1对应的海明码，其它以此类推……</p>
</div>
<p>通过上面的计算，我们就可以得到一个经过编码之后的数据：0110 0111 1001</p>
<table>
<thead>
<tr>
<th style="text-align:center">位置</th>
<th style="text-align:center">12</th>
<th style="text-align:center">11</th>
<th style="text-align:center">10</th>
<th style="text-align:center">9</th>
<th style="text-align:center">8</th>
<th style="text-align:center">7</th>
<th style="text-align:center">6</th>
<th style="text-align:center">5</th>
<th style="text-align:center">4</th>
<th style="text-align:center">3</th>
<th style="text-align:center">2</th>
<th style="text-align:center">1</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:center">最终信息位</td>
<td style="text-align:center">0</td>
<td style="text-align:center">1</td>
<td style="text-align:center">1</td>
<td style="text-align:center">0</td>
<td style="text-align:center">0</td>
<td style="text-align:center">1</td>
<td style="text-align:center">1</td>
<td style="text-align:center">1</td>
<td style="text-align:center">1</td>
<td style="text-align:center">0</td>
<td style="text-align:center">0</td>
<td style="text-align:center">1</td>
</tr>
</tbody>
</table>
<p>接收端收到对应的数据之后，开始校验数据是否有错误</p>
<p>假设 0110 0111 1001 ->传输为 011<span class='p red'>1</span> 0111 1001</p>
<p>继续根据上面的方式，重新计算H1，H2，H4，H8校验位，计算的结果为<span class='p cyan'>1100</span></p>
<p>然后拿之前的校验码和新算出来的校验码求异或0101 ⊕ <span class='p cyan'>1100</span> = 1001 = 9，所以可以判断出是第9位出现了错误，修改之后就完成了纠错</p>
<div class="note success no-icon flat"><p>以上就是海明码检错以及纠错的过程</p>
</div>
<hr>
<div class="note warning flat"><p>如果你认真的看到这里，你一定会有一个疑问，如果出现两个以上的错误时，还能纠错吗？下面实践一下</p>
</div>
<h2 id="栗子2🌰">栗子2🌰</h2>
<p>假设出现了两个数据位的错误 0110 0111 1001 ->传输为 01<span class='p red'>01</span> 0111 1001</p>
<table>
<thead>
<tr>
<th style="text-align:center">位置</th>
<th style="text-align:center">12</th>
<th style="text-align:center">11</th>
<th style="text-align:center">10</th>
<th style="text-align:center">9</th>
<th style="text-align:center">8</th>
<th style="text-align:center">7</th>
<th style="text-align:center">6</th>
<th style="text-align:center">5</th>
<th style="text-align:center">4</th>
<th style="text-align:center">3</th>
<th style="text-align:center">2</th>
<th style="text-align:center">1</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:center">最终信息位</td>
<td style="text-align:center">0</td>
<td style="text-align:center">1</td>
<td style="text-align:center"><span class='p red'>0</span></td>
<td style="text-align:center"><span class='p red'>1</span></td>
<td style="text-align:center">0</td>
<td style="text-align:center">1</td>
<td style="text-align:center">1</td>
<td style="text-align:center">1</td>
<td style="text-align:center">1</td>
<td style="text-align:center">0</td>
<td style="text-align:center">0</td>
<td style="text-align:center">1</td>
</tr>
</tbody>
</table>
<p>再次计算校验位H1，H2，H4，H8为<span class='p cyan'>0110</span><br>
再次与之前的校验码求异或 0101 ⊕ <span class='p cyan'>0110</span> = 0011 = 3<br>
按照这次的计算结果来看，是第三位数据出现了问题，但是很明显我们出错的是第9,10位，但是，当你把第三位修改之后，再次进行校验，发现没有问题了😗</p>
<div class="note danger no-icon flat"><p>所以通过上面的第二个例子我们可以直到，海明码只能检测一位错误，当出现两个以上的错误是，纠错结果会变得不准确</p>
</div>
<h1>最终总结</h1>
<table>
<thead>
<tr>
<th style="text-align:left">特点</th>
<th style="text-align:left">奇偶校验</th>
<th style="text-align:left">CRC</th>
<th style="text-align:left">海明码</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left">原理</td>
<td style="text-align:left">在数据中添加一个额外的校验位，使整个数据的1的个数为奇数（奇校验）或偶数（偶校验），用于检测单个比特的错误。</td>
<td style="text-align:left">利用除法操作，在发送的数据后面附加一个固定长度的校验码（CRC码），接收端通过相同的除法操作验证数据的完整性。</td>
<td style="text-align:left">通过在数据中添加冗余位（校验位），使得整个码字中特定位置的位数（1的个数）具有确定的奇偶性，以便检测和纠正错误。</td>
</tr>
<tr>
<td style="text-align:left">错误检测能力</td>
<td style="text-align:left">只能检测单个比特的错误，对于两个或更多比特的错误可能无法检测。</td>
<td style="text-align:left">能够检测并纠正多个比特的错误，具体取决于CRC码的长度。</td>
<td style="text-align:left">可以检测并纠正单个比特的错误，有时还能检测并纠正两个比特的错误。</td>
</tr>
<tr>
<td style="text-align:left">错误纠正能力</td>
<td style="text-align:left">无错误纠正能力。</td>
<td style="text-align:left">根据CRC码的长度和算法设计，可以纠正一定数量的错误。</td>
<td style="text-align:left">可以纠正单个比特的错误。</td>
</tr>
<tr>
<td style="text-align:left">开销</td>
<td style="text-align:left">仅需添加一个校验位，开销最小。</td>
<td style="text-align:left">需要添加固定长度的CRC码，开销适中。</td>
<td style="text-align:left">需要添加多个校验位，开销相对较大。</td>
</tr>
<tr>
<td style="text-align:left">实现复杂度</td>
<td style="text-align:left">最简单，只需进行简单的位运算。</td>
<td style="text-align:left">中等复杂度，需要选择合适的CRC多项式和实现除法操作。</td>
<td style="text-align:left">相对较高，需要设计合适的校验位位置和计算校验位的值。</td>
</tr>
<tr>
<td style="text-align:left">应用场景</td>
<td style="text-align:left">常用于简单的数据传输和存储系统，对错误检测和纠正要求不高的场合。</td>
<td style="text-align:left">广泛应用于网络通信、数据存储等领域，用于确保数据的完整性。</td>
<td style="text-align:left">适用于对错误检测和纠正有较高要求的应用，如内存错误检测。</td>
</tr>
</tbody>
</table>
<div class="note purple icon-padding disabled"><i class="note-icon far fa-hand-scissors"></i><p>笔记到这里就结束啦，你学会了么😮</p>
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class="author-info__bottom-group"><a class="author-info__bottom-group-left" href="/"><h1 class="author-info__name">伴随</h1><div class="author-info__desc">请保持那份热爱 奔赴下一场山海</div></a><div class="card-info-social-icons is-center"><a class="social-icon faa-parent animated-hover" href="https://qm.qq.com/q/1sv8urqpRS" target="_blank" title="QQ"><i class="anzhiyufont anzhiyu-icon-qq"></i></a></div></div></div></div><div class="card-widget card-announcement"><div class="item-headline"><i class="anzhiyufont anzhiyu-icon-bullhorn anzhiyu-shake"></i><span>公告</span></div><div class="announcement_content"><div id="welcome-info"></div></div></div><div class="sticky_layout"><div class="card-widget" id="card-toc"><div class="item-headline"><i class="anzhiyufont anzhiyu-icon-bars"></i><span>文章目录</span><span class="toc-percentage"></span></div><div class="toc-content"><ol class="toc"><li class="toc-item toc-level-1"><a class="toc-link"><span class="toc-number">1.</span> <span class="toc-text">1. 为什么要有校验码？</span></a></li><li class="toc-item toc-level-1"><a class="toc-link"><span class="toc-number">2.</span> <span class="toc-text">2. 奇偶校验</span></a><ol class="toc-child"><li class="toc-item toc-level-2"><a class="toc-link" href="#%E6%A0%A1%E9%AA%8C%E4%BD%8D%E7%9A%84%E8%AE%A1%E7%AE%97"><span class="toc-number">2.1.</span> <span class="toc-text">校验位的计算</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E4%B8%BE%E4%B8%AA%E6%A0%97%E5%AD%90%F0%9F%8C%B0"><span class="toc-number">2.2.</span> <span class="toc-text">举个栗子🌰</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E6%80%BB%E7%BB%93"><span class="toc-number">2.3.</span> <span class="toc-text">总结</span></a></li></ol></li><li class="toc-item toc-level-1"><a class="toc-link"><span class="toc-number">3.</span> <span class="toc-text">2. CRC（循环冗余校验）</span></a><ol class="toc-child"><li class="toc-item toc-level-2"><a class="toc-link" href="#%E6%A0%A1%E9%AA%8C%E4%BD%8D%E7%9A%84%E8%AE%A1%E7%AE%97-v2"><span class="toc-number">3.1.</span> <span class="toc-text">校验位的计算</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E4%B8%BE%E4%B8%AA%E6%A0%97%E5%AD%90%F0%9F%8C%B0-v2"><span class="toc-number">3.2.</span> <span class="toc-text">举个栗子🌰</span></a></li></ol></li><li class="toc-item toc-level-1"><a class="toc-link"><span class="toc-number">4.</span> <span class="toc-text">3. 海明校验码</span></a><ol class="toc-child"><li class="toc-item toc-level-2"><a class="toc-link" href="#%E6%A0%A1%E9%AA%8C%E4%BD%8D%E4%B8%8E%E6%95%B0%E6%8D%AE%E4%BD%8D%E7%9A%84%E5%85%B3%E7%B3%BB"><span class="toc-number">4.1.</span> <span class="toc-text">校验位与数据位的关系</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E4%B8%BE%E4%B8%AA%E6%A0%97%E5%AD%90%F0%9F%8C%B0-v3"><span class="toc-number">4.2.</span> <span class="toc-text">举个栗子🌰</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E6%A0%97%E5%AD%902%F0%9F%8C%B0"><span class="toc-number">4.3.</span> <span class="toc-text">栗子2🌰</span></a></li></ol></li><li class="toc-item toc-level-1"><a class="toc-link"><span class="toc-number">5.</span> <span class="toc-text">最终总结</span></a></li></ol></div></div><div class="card-widget card-recent-post"><div class="item-headline"><i class="anzhiyufont anzhiyu-icon-history"></i><span>最近发布</span></div><div class="aside-list"><div class="aside-list-item"><a class="thumbnail" href="/7ac.html" title="JAVA简易日志工具类"><img src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" onerror="this.onerror=null,this.src="https://up.ctext.top/default_bg.png"" data-lazy-src="https://up.ctext.top/article/2025/11/log.png" onerror="this.onerror=null;this.src='https://up.ctext.top/default_bg.png'" alt="JAVA简易日志工具类"/></a><div class="content"><a class="title" href="/7ac.html" 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img.src = "https://up.ctext.top/happy.svg";
img.title = "下班了就该开开心心的玩耍，嘿嘿~";
img.alt = "下班了就该开开心心的玩耍，嘿嘿~";
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nick: '你的专属昵称',
nickError: '昵称太短啦！至少得有3个字符才能算数哦！',
mail: '你的电子小信箱',
mailError: '哎呀，这不是一个合格的邮箱地址哦，请再检查一遍！',
link: '你有个酷酷的网址吗？',
optional: '这个嘛，想填就填，不填也行！',
placeholder: '吐槽一下？',
sofa: '来，坐沙发上，发表你的评论吧！',
submit: '一键发送，让世界听到你的声音！',
like: '点个赞，让TA知道你的心意！',
cancelLike: '哎呀，不小心点错了，取消赞！',
reply: '回复TA，开启一场对话吧！',
cancelReply: '等等，我话还没说完，取消回复！',
comment: '来，说说你的看法，让评论区热闹起来！',
refresh: '刷新一下，看看有啥新动态！',
more: '哇塞，还有更多精彩内容等你来加载！',
preview: '先睹为快，预览一下你的大作吧！',
emoji: '来，加点表情，让你的评论更生动！',
uploadImage: '上传你的美照，让大家一起欣赏！',
seconds: '刚刚才发的，新鲜出炉！',
minutes: '几分钟前，还热乎着呢！',
hours: '几小时前，热度还在哦！',
days: '几天前，但依旧值得一看！',
now: '哇塞，刚刚发的，快抢沙发！',
uploading: '稍等片刻，你的图片正在飞奔而来！',
login: '登录一下，解锁更多功能！',
logout: '拜拜了您嘞，下次再见！',
admin: '博主大人在此，谁敢造次！',
sticky: '置顶一下，让更多人看到你的精彩内容！',
word: '字',
wordHint: '评论字数要在0到1字之间哦！\n现在你已经写了$2个字了，加油！',
anonymous: '神秘人出没，猜猜是谁？',
level0: '潜水员，深藏不露！',
level1: '冒泡小能手，偶尔露个头！',
level2: '吐槽大师，一针见血！',
level3: '活跃分子，无处不在！',
level4: '话痨达人，说个不停！',
level5: '传说级大神，膜拜一下！',
gif: '动态表情包，让你的评论更有趣！',
gifSearchPlaceholder: '快来搜索你喜欢的表情包，让评论更生动！',
profile: '看看我的个人资料，了解一下我吧！',
approved: '审核通过，可以展示啦！',
waiting: '正在等待审核，别急哦！',
spam: '垃圾内容，走开走开！',
unsticky: '换个位置看看！',
oldest: '看看最早的评论！',
latest: '看看最新的动态！',
hottest: '看看哪条评论最火！',
reactionTitle: '快来对这篇文章发表你的看法吧，让博主知道你的想法！😜'
};
let initFn = window.walineFn || null
const initWaline = (Fn) => {
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serverURL: 'https://bc.ctext.top/',
pageview: false,
dark: 'html[data-theme="dark"]',
path: window.location.pathname,
comment: false,
imageUploader: true,
comment: false,
copyright: false,
reaction: true,
locale
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waline.destroy()
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anzhiyu.addGlobalFn('pjax', destroyWaline, 'destroyWaline')
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const { init } = await import('https://cdn.cbd.int/@waline/client@3.1.3/dist/waline.js')
initFn = init || Waline.init
initWaline(initFn)
window.walineFn = initFn
}
}
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content = content.replace(/<pre><code>.*?<\/pre>/gi, '[代码]') // replace code
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generateHtml(walineArray)
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