试证明[tex=10.929x1.357]QAILzNsNNiyt0WYGmZvAmHfiTP5pnEt0AgkfJkFdk4JJnJWqm3v0ZcLJCaUrhtxh[/tex] 为[tex=2.571x1.357]l4S8rNYfAlihcTRikoK2rQ==[/tex] 循环码的生成多项式。求出此循环码的生成矩阵,并写出消息码为[tex=6.643x1.5]7/qRAMyi3xWKeI1Mgkd6bMyDzdTm1JzcfjFvN7Ua90U=[/tex] 时的码多项式(系统码形式)。

2022-06-09

试证明[tex=10.929x1.357]QAILzNsNNiyt0WYGmZvAmHfiTP5pnEt0AgkfJkFdk4JJnJWqm3v0ZcLJCaUrhtxh[/tex] 为[tex=2.571x1.357]l4S8rNYfAlihcTRikoK2rQ==[/tex] 循环码的生成多项式。求出此循环码的生成矩阵,并写出消息码为[tex=6.643x1.5]7/qRAMyi3xWKeI1Mgkd6bMyDzdTm1JzcfjFvN7Ua90U=[/tex] 时的码多项式(系统码形式)。

答案：

分析:若要证明此多项式为生成多项式,只需证明它符合习题[tex=2.286x1.143]rpBfOEfWerJIjbmLhGMyKw==[/tex]列出的生成多项式满足的三个条件。 （1）证明 设 [tex=13.5x1.5]q5cTZN25dWUlEPKW60zrpxs4zYOssNgRKgbt6YqfsyNB9cnxwFgmlJZSWvhaVHni[/tex]。① [tex=1.857x1.357]fBOYuAIZ/H4m1Dx+my86tg==[/tex]的最高次赛为 10 ,符合[tex=1.857x1.357]fBOYuAIZ/H4m1Dx+my86tg==[/tex]是惟一一个[tex=7.571x1.357]qAH+9C1SZHgZw+SoEvVsvA==[/tex]次多项式的条件;② [tex=1.857x1.357]fBOYuAIZ/H4m1Dx+my86tg==[/tex]的常数项不为 0 ;③ 作 [tex=9.143x2.786]Qq5oQPNAj9tDHUQNgcO4LkVzMLk29ZUr5kIJ/ojWWofR6dv6GMvdHrMtIUUKJM1Q[/tex], 可见 [tex=2.643x1.357]dwjVzTk674fu9agMhHS2Lw==[/tex]可以被[tex=1.643x1.357]dVY93ll8VMQ0iaNbw05kQQ==[/tex] 整除(余数为 0 ),表示 [tex=1.857x1.357]fBOYuAIZ/H4m1Dx+my86tg==[/tex] 是 [tex=2.643x1.357]dwjVzTk674fu9agMhHS2Lw==[/tex]的一个因式。由以上三条得证， [tex=1.857x1.357]fBOYuAIZ/H4m1Dx+my86tg==[/tex]为[tex=2.571x1.357]CuLbxWg9vVfXExpMeAkdQw==[/tex]循环码的生成多项式。(2)该 [tex=2.571x1.357]l4S8rNYfAlihcTRikoK2rQ==[/tex]循环码的生成矩阵为[tex=23.714x7.786]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[/tex]或  [tex=25.714x5.786]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[/tex](3) 码多项式的系统码形式为[tex=9.643x1.5]3LNUuda/I43sxIpCHi5vjLkZ8htfkgKsrb7L+X6LaoQ=[/tex]式中: [tex=1.857x1.357]AWAXM5J6sQC9QzjXrKoUjw==[/tex] 是 [tex=6.5x1.5]y3hY+IkkHE1PqAVo6l6WnyBOLDmp9XnKKLNkUQDizFA=[/tex]的余式。因此,做以下运算:[tex=28.929x5.929]qeiYnKXLEhyhuGRg8yLtr19yKu5SIdOnGgHLvF5EH0jiA653S2Yx/FTJvFe27BfIfBu4ttRvqI7MTtaUM/ROcrA1M2rGdc7fwlNXqlPkuVZ/g9OSvM5ocUHc39lTeTB1ugH0BgVLig+2KF/1HirJPDSLiRyd1BIE14GipKa6SMzIzHXgY5LXbPmjq9oWXCavOVMAR90Tf4Mzv3+vDMVr2D62VFnHHjc6FfhqiWbasz+tL7c+Ci/vqhfjiaWvzilEAYPagd0N+uoJF34H3RO/cTeoR6P1CSM/4GTCxD7sBVbsp/rv2066K4jFqUPD+Xqgs+svU8roOktkuohaqR9EVt+9laGr8wknGR7jS+3EUaxx4RyG4nN1F5BQgO2zD4Hq[/tex]得到余式[tex=8.357x1.5]cNdH6BWzV2pHj/xJrIykC988GdXJ4/rDdjFDsdru3AY=[/tex]则信息码[tex=6.643x1.5]7/qRAMyi3xWKeI1Mgkd6bMyDzdTm1JzcfjFvN7Ua90U=[/tex]所对应的码多项式为[tex=23.357x3.357]qeiYnKXLEhyhuGRg8yLtr11+9E01GToPkvcW7tDB6Y4Rnr89n2rIi36bUTBLXVp2GqL2CsGCTgdxh+Fv4tEfNU3VHSN8xqlHlHn1LE35SUWdFXxyDWq0VE3jPR0j457bc6fyMJ9Ge0E2/uGMdJLtFj+d4S+SB3g5ukjlJVmhX14gezkiBAeTbxJ9dxcD+UKcVqoMF9w2htBvCv1t84N6VQ==[/tex]所对应的码组为 [tex=7.5x1.0]cuz+Vbi8JNJ5ZHxt/QV6vA==[/tex]

举一反三

内容

0
已知[tex=1.714x1.357]GtGi/bgZIEmseV5BIOBL9g==[/tex])循环码的生成多项式为[tex=6.643x1.5]h6XIQiULBKNc9dBUvH0Ak4mvj0HpijEEJdGMTDFLhqw=[/tex]当信息码组为[tex=2.0x1.0]YG5gO8sL0Tpqa51D7u15mw==[/tex],求编码后的循环码组
1
已知(7,4)循环码的生成多项式为 [tex=6.214x1.5]ajhTM+YYfvaFCB+YvrFXh2e92QY7OzIEJNEhAkI/sU8=[/tex] 若信息码为 0110, 编出相应的码字(系统码)
2
. 已知[tex=2.071x1.357]fKtHfy5PkM/VRAijZ52i8A==[/tex] 循环码的生成矩阵为[tex=13.5x3.5]Sn9tiYOlmZ7h9vHEJ7cBWz6hYtN6pEQebQDCS/1hZS3SY2Gq+CDlOuW7LI5lnUMvdlYxn/gqzsH5CHc+KUq912S8kbbM2JI9Oz1ccuos+IYjUZN6Go/EhRh7qvb3lGN9sxTWOGfovxNx6we5NUwOTW07h2TixaMYpX0tovBxsseDfnFtSVjfV0xwTIPAAOTqG8FEXMz6ajMu3obo0aWKkw==[/tex]试写出该码的生成多项式 [tex=1.857x1.357]fBOYuAIZ/H4m1Dx+my86tg==[/tex]和监督矩阵[tex=1.0x1.0]5cCYlc8KT4YbU5w753RaBuepE+1r6xIU/VfSPN0Fj+0=[/tex]
3
已知[tex=2.071x1.357]fKtHfy5PkM/VRAijZ52i8A==[/tex] 循环码的生成矩阵为[tex=7.429x3.5]krqLcez/NatiMpfHupaHv/jBV+daUeIEAvV3HGP7/6NJt7XUAYPw293TN4CfM3zjQyjckZx/nkC6ZffesezmeOiosuySiPKQuLl2ckZaZ0k=[/tex]若输入信息码为 [tex=1.5x1.0]SYPwVg8E46efdwmIbbbt2Q==[/tex] ,试写出对应的循环码码组;[br][/br]
4
已知一种[tex=2.071x1.357]fKtHfy5PkM/VRAijZ52i8A==[/tex] 循环码的全部码组为 [tex=18.0x2.357]CeOWlpLvH8Qhk/RmfIvBHYkVEkPOY8MuFlzJuPaF9YRDXiumO8Aw9HMXJ3FztJm0XTzE9WAIMhp/rwMwn0zTjQRGhMGd/A9A19vIj15rSms1FDLaa5aKyX8sP3SLQ3hfvZbUz3wtAMuHcY7Uc5OTtg==[/tex]试求: 该循环码的生成多项式[tex=1.857x1.357]fBOYuAIZ/H4m1Dx+my86tg==[/tex] 、典型生成矩阵[tex=0.929x1.0]rGq0+wwusEFO0LgjMJZJMvpknGyDqi+aORVwkF+3ikY=[/tex]和典型监督矩阵 [tex=1.286x1.214]Xw+l023EljiREVAQxLXBUYgCueuo+JmPgja9qBvB5/4=[/tex]