{"id":97959,"date":"2026-08-30T20:56:44","date_gmt":"2026-08-30T12:56:44","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/97959.html"},"modified":"2026-08-30T20:56:44","modified_gmt":"2026-08-30T12:56:44","slug":"%e6%b5%b7%e6%98%8e%e7%a0%81%ef%bc%9a%e4%bb%8e%e7%bc%96%e7%a0%81%e5%88%b0%e7%ba%a0%e9%94%99%ef%bc%8c%e4%b8%80%e7%af%87%e8%ae%b2%e9%80%8f","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/97959.html","title":{"rendered":"\u6d77\u660e\u7801\uff1a\u4ece\u7f16\u7801\u5230\u7ea0\u9519\uff0c\u4e00\u7bc7\u8bb2\u900f"},"content":{"rendered":"<\/p>\n<h4>\u6587\u7ae0\u76ee\u5f55<\/h4>\n<ul>\n<li>\n<ul>\n<li>\n<ul>\n<li>\u6d77\u660e\u7801&#xff1a;\u4ece\u7f16\u7801\u5230\u7ea0\u9519&#xff0c;\u4e00\u7bc7\u8bb2\u900f<\/li>\n<li>\u4e00\u3001\u786e\u5b9a\u6821\u9a8c\u4f4d\u6570\u91cf<\/li>\n<li>\u4e8c\u3001\u7f16\u7801\u8fc7\u7a0b<\/li>\n<li>\n<ul>\n<li>\u7b2c 1 \u6b65&#xff1a;\u786e\u5b9a\u6821\u9a8c\u4f4d\u548c\u6570\u636e\u4f4d\u7684\u4f4d\u7f6e<\/li>\n<li>\u7b2c 2 \u6b65&#xff1a;\u786e\u5b9a\u6bcf\u4e2a\u6821\u9a8c\u4f4d\u8d1f\u8d23\u68c0\u67e5\u54ea\u4e9b\u4f4d<\/li>\n<li>\u7b2c 3 \u6b65&#xff1a;\u8ba1\u7b97\u6bcf\u4e2a\u6821\u9a8c\u4f4d\u7684\u503c&#xff08;\u5076\u6821\u9a8c&#xff09;<\/li>\n<li>\u7b2c 4 \u6b65&#xff1a;\u5f97\u5230\u6700\u7ec8\u7f16\u7801<\/li>\n<\/ul>\n<\/li>\n<li>\u4e09\u3001\u7ea0\u9519\u8fc7\u7a0b<\/li>\n<li>\n<ul>\n<li>\u7b2c 1 \u6b65&#xff1a;\u91cd\u65b0\u8ba1\u7b97\u6bcf\u4e2a\u6821\u9a8c\u7ec4\u7684\u5947\u5076\u6027<\/li>\n<li>\u7b2c 2 \u6b65&#xff1a;\u62fc\u51fa\u9519\u8bef\u4f4d\u7f6e\u7f16\u53f7<\/li>\n<li>\u7b2c 3 \u6b65&#xff1a;\u7ea0\u6b63\u9519\u8bef<\/li>\n<\/ul>\n<\/li>\n<li>\u56db\u3001\u5982\u679c\u6ca1\u6709\u9519\u8bef\u5462&#xff1f;<\/li>\n<li>\u4e94\u3001\u603b\u7ed3<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h4>\u6d77\u660e\u7801&#xff1a;\u4ece\u7f16\u7801\u5230\u7ea0\u9519&#xff0c;\u4e00\u7bc7\u8bb2\u900f<\/h4>\n<p>\u6d77\u660e\u7801&#xff08;Hamming Code&#xff09;\u662f\u7ea0\u9519\u7f16\u7801\u4e2d\u6700\u7ecf\u5178\u3001\u6700\u57fa\u7840\u7684\u65b9\u6848&#xff0c;\u7531\u7406\u67e5\u5fb7\u00b7\u6d77\u660e\u4e8e 1950 \u5e74\u63d0\u51fa\u3002\u5b83\u7684\u6838\u5fc3\u80fd\u529b\u662f&#xff1a;\u81ea\u52a8\u68c0\u6d4b\u5e76\u7ea0\u6b63 1 \u4f4d\u9519\u8bef\u3002<\/p>\n<p>\u4eca\u5929\u6211\u4eec\u5c31\u7528 4 \u4f4d\u6570\u636e 1011 \u4f5c\u4e3a\u4f8b\u5b50&#xff0c;\u5b8c\u6574\u8d70\u4e00\u904d\u6d77\u660e\u7801\u7684\u7f16\u7801\u548c\u7ea0\u9519\u8fc7\u7a0b\u3002<\/p>\n<hr \/>\n<h4>\u4e00\u3001\u786e\u5b9a\u6821\u9a8c\u4f4d\u6570\u91cf<\/h4>\n<p>\u6821\u9a8c\u4f4d\u6570\u91cf r \u9700\u6ee1\u8db3&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>          2<\/p>\n<p>          r<\/p>\n<p>         \u2265<\/p>\n<p>         d<\/p>\n<p>         &#043;<\/p>\n<p>         r<\/p>\n<p>         &#043;<\/p>\n<p>         1<\/p>\n<p>        2^r \\\\geq d &#043; r &#043; 1<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8504em;vertical-align: -0.136em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7144em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0278em\">r<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2265<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7778em;vertical-align: -0.0833em\"><\/span><span class=\"mord mathnormal\">d<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6667em;vertical-align: -0.0833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">r<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u5176\u4e2d d \u4e3a\u6570\u636e\u4f4d\u6570\u3002<\/p>\n<p>\u5bf9\u4e8e 4 \u4f4d\u6570\u636e&#xff1a;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         2<\/p>\n<p>         3<\/p>\n<p>        &#061;<\/p>\n<p>        8<\/p>\n<p>        \u2265<\/p>\n<p>        4<\/p>\n<p>        &#043;<\/p>\n<p>        3<\/p>\n<p>        &#043;<\/p>\n<p>        1<\/p>\n<p>        &#061;<\/p>\n<p>        8<\/p>\n<p>       2^3 &#061; 8 \\\\geq 4 &#043; 3 &#043; 1 &#061; 8<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8141em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7804em;vertical-align: -0.136em\"><\/span><span class=\"mord\">8<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2265<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7278em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">4<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7278em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">3<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">8<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u6240\u4ee5 r &#061; 3\u3002<\/p>\n<p>\u603b\u5171 7 \u4f4d\u7f16\u7801&#xff0c;\u5373 (7, 4) \u6d77\u660e\u7801\u3002<\/p>\n<hr \/>\n<h4>\u4e8c\u3001\u7f16\u7801\u8fc7\u7a0b<\/h4>\n<h5>\u7b2c 1 \u6b65&#xff1a;\u786e\u5b9a\u6821\u9a8c\u4f4d\u548c\u6570\u636e\u4f4d\u7684\u4f4d\u7f6e<\/h5>\n<p>\u6821\u9a8c\u4f4d\u56fa\u5b9a\u653e\u5728\u4f4d\u7f6e\u7f16\u53f7\u4e3a 2 \u7684\u5e42\u6b21\u7684\u4f4d\u4e0a&#xff0c;\u5176\u4f59\u4f4d\u7f6e\u653e\u6570\u636e\u4f4d&#xff1a;<\/p>\n<table>\n<tr>\u4f4d\u7f6e1234567<\/tr>\n<tbody>\n<tr>\n<td>\u7c7b\u578b<\/td>\n<td>P\u2081<\/td>\n<td>P\u2082<\/td>\n<td>D\u2081<\/td>\n<td>P\u2083<\/td>\n<td>D\u2082<\/td>\n<td>D\u2083<\/td>\n<td>D\u2084<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ul>\n<li>P &#061; \u6821\u9a8c\u4f4d&#xff08;\u7b2c 1\u30012\u30014 \u4f4d&#xff09;<\/li>\n<li>D &#061; \u6570\u636e\u4f4d&#xff08;\u7b2c 3\u30015\u30016\u30017 \u4f4d&#xff09;<\/li>\n<\/ul>\n<p>\u5c06\u6570\u636e 1011 \u4f9d\u6b21\u586b\u5165\u6570\u636e\u4f4d&#xff1a;<\/p>\n<table>\n<tr>\u4f4d\u7f6e1234567<\/tr>\n<tbody>\n<tr>\n<td>\u7c7b\u578b<\/td>\n<td>P\u2081<\/td>\n<td>P\u2082<\/td>\n<td>D\u2081<\/td>\n<td>P\u2083<\/td>\n<td>D\u2082<\/td>\n<td>D\u2083<\/td>\n<td>D\u2084<\/td>\n<\/tr>\n<tr>\n<td>\u503c<\/td>\n<td>?<\/td>\n<td>?<\/td>\n<td>1<\/td>\n<td>?<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h5>\u7b2c 2 \u6b65&#xff1a;\u786e\u5b9a\u6bcf\u4e2a\u6821\u9a8c\u4f4d\u8d1f\u8d23\u68c0\u67e5\u54ea\u4e9b\u4f4d<\/h5>\n<p>\u89c4\u5219&#xff1a;\u4f4d\u7f6e\u7f16\u53f7\u7684\u4e8c\u8fdb\u5236\u8868\u793a\u4e2d&#xff0c;\u7b2c i \u4f4d\u4e3a 1 \u7684\u90a3\u4e9b\u4f4d\u7f6e&#xff0c;\u5f52 P\u1d62 \u68c0\u67e5\u3002<\/p>\n<ul>\n<li>P\u2081&#xff08;\u7b2c 1 \u4f4d&#xff09;&#xff1a;\u68c0\u67e5\u4f4d\u7f6e\u7f16\u53f7\u4e8c\u8fdb\u5236\u672b\u4f4d\u4e3a 1 \u7684\u4f4d \u2192 \u7b2c 1\u30013\u30015\u30017 \u4f4d<\/li>\n<li>P\u2082&#xff08;\u7b2c 2 \u4f4d&#xff09;&#xff1a;\u68c0\u67e5\u4f4d\u7f6e\u7f16\u53f7\u4e8c\u8fdb\u5236\u5012\u6570\u7b2c 2 \u4f4d\u4e3a 1 \u7684\u4f4d \u2192 \u7b2c 2\u30013\u30016\u30017 \u4f4d<\/li>\n<li>P\u2083&#xff08;\u7b2c 4 \u4f4d&#xff09;&#xff1a;\u68c0\u67e5\u4f4d\u7f6e\u7f16\u53f7\u4e8c\u8fdb\u5236\u5012\u6570\u7b2c 3 \u4f4d\u4e3a 1 \u7684\u4f4d \u2192 \u7b2c 4\u30015\u30016\u30017 \u4f4d<\/li>\n<\/ul>\n<h5>\u7b2c 3 \u6b65&#xff1a;\u8ba1\u7b97\u6bcf\u4e2a\u6821\u9a8c\u4f4d\u7684\u503c&#xff08;\u5076\u6821\u9a8c&#xff09;<\/h5>\n<p>\u8ba9\u6bcf\u4e2a\u6821\u9a8c\u4f4d\u8d1f\u8d23\u7684\u90a3\u4e9b\u4f4d\u4e2d&#xff0c;1 \u7684\u4e2a\u6570\u4e3a\u5076\u6570&#xff1a;<\/p>\n<p>P\u2081&#xff1a;\u68c0\u67e5\u7b2c 1\u30013\u30015\u30017 \u4f4d \u2192 P\u2081 &#043; D\u2081 &#043; D\u2082 &#043; D\u2084 &#061; P\u2081 &#043; 1 &#043; 0 &#043; 1 &#061; P\u2081 &#043; 2 \u8981\u4f7f\u603b\u548c\u4e3a\u5076\u6570 \u2192 P\u2081 &#061; 0<\/p>\n<p>P\u2082&#xff1a;\u68c0\u67e5\u7b2c 2\u30013\u30016\u30017 \u4f4d \u2192 P\u2082 &#043; D\u2081 &#043; D\u2083 &#043; D\u2084 &#061; P\u2082 &#043; 1 &#043; 1 &#043; 1 &#061; P\u2082 &#043; 3 \u8981\u4f7f\u603b\u548c\u4e3a\u5076\u6570 \u2192 P\u2082 &#061; 1<\/p>\n<p>P\u2083&#xff1a;\u68c0\u67e5\u7b2c 4\u30015\u30016\u30017 \u4f4d \u2192 P\u2083 &#043; D\u2082 &#043; D\u2083 &#043; D\u2084 &#061; P\u2083 &#043; 0 &#043; 1 &#043; 1 &#061; P\u2083 &#043; 2 \u8981\u4f7f\u603b\u548c\u4e3a\u5076\u6570 \u2192 P\u2083 &#061; 0<\/p>\n<h5>\u7b2c 4 \u6b65&#xff1a;\u5f97\u5230\u6700\u7ec8\u7f16\u7801<\/h5>\n<table>\n<tr>\u4f4d\u7f6e1234567<\/tr>\n<tbody>\n<tr>\n<td>\u7c7b\u578b<\/td>\n<td>P\u2081<\/td>\n<td>P\u2082<\/td>\n<td>D\u2081<\/td>\n<td>P\u2083<\/td>\n<td>D\u2082<\/td>\n<td>D\u2083<\/td>\n<td>D\u2084<\/td>\n<\/tr>\n<tr>\n<td>\u503c<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>1<\/td>\n<td>0<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u6700\u7ec8\u7f16\u7801\u4e3a&#xff1a;0110011<\/p>\n<hr \/>\n<h4>\u4e09\u3001\u7ea0\u9519\u8fc7\u7a0b<\/h4>\n<p>\u5047\u8bbe\u4f20\u8f93\u8fc7\u7a0b\u4e2d\u7b2c 5 \u4f4d\u51fa\u9519&#xff0c;\u63a5\u6536\u65b9\u6536\u5230\u7684\u7f16\u7801\u4e3a&#xff1a;<\/p>\n<p>0 1 1 0 1 1 1<\/p>\n<p>&#xff08;\u7b2c 5 \u4f4d\u4ece 0 \u53d8\u6210\u4e86 1&#xff09;<\/p>\n<h5>\u7b2c 1 \u6b65&#xff1a;\u91cd\u65b0\u8ba1\u7b97\u6bcf\u4e2a\u6821\u9a8c\u7ec4\u7684\u5947\u5076\u6027<\/h5>\n<p>P\u2081 \u7ec4&#xff08;\u7b2c 1\u30013\u30015\u30017 \u4f4d&#xff09;&#xff1a;0 &#043; 1 &#043; 1 &#043; 1 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