查看“︁对偶码”︁的源代码

在[[编码理论]]中，{{le|线性码|linear code}}的'''对偶码'''（{{lang-en|'''dual code'''}}）

:<math>C\subset\mathbb{F}_q^n</math>

是有如下定义

:<math>C^\perp = \{x \in \mathbb{F}_q^n \mid \langle x,c\rangle = 0\;\forall c \in C \} </math>

的线性码，其中

:<math>\langle x, c \rangle = \sum_{i=1}^n x_i c_i </math>

是一个数量积。用[[线性代数]]的属于来说，对偶码是 ''C'' 对[[雙線性形式]] <,> 的{{le|零化子 (环论)|Annihilator (ring theory)|零化子}}。 ''C'' 的[[向量空间的维数|维数]]与其对偶的维数加起来总为长度 ''n''：

:<math>\dim C + \dim C^\perp = n.</math>

对偶码的[[生成矩阵]]是原码的[[奇偶檢驗矩陣]]，反之亦然。对偶码的对偶总是原码。

==参考文献==
{{reflist}}
{{refbegin}}
* {{cite book | last=Hill | first=Raymond | title=A first course in coding theory | url=https://archive.org/details/firstcourseincod0000hill | publisher=[[牛津大學出版社|Oxford University Press]] | series=Oxford Applied Mathematics and Computing Science Series | date=1986 | isbn=0-19-853803-0 | page=[https://archive.org/details/firstcourseincod0000hill/page/67 67] }}
* {{cite book | last = Pless | first = Vera | authorlink=Vera Pless | title = Introduction to the theory of error-correcting codes | url = https://archive.org/details/introductiontoth0000ples_o1b9 | publisher = [[約翰威立|John Wiley & Sons]]|series = Wiley-Interscience Series in Discrete Mathematics | date = 1982| isbn = 0-471-08684-3 | page=[https://archive.org/details/introductiontoth0000ples_o1b9/page/8 8] }}
* {{cite book | author=J.H. van Lint | title=Introduction to Coding Theory | edition=2nd | publisher=Springer-Verlag | series=[[數學研究生教材|GTM]] | volume=86 | date=1992 | isbn=3-540-54894-7 | page=34}}
{{refend}}

== 外部链接 ==
* [https://web.archive.org/web/20170215050646/http://www.maths.manchester.ac.uk/~pas/code/notes/part9.pdf MATH32031: Coding Theory - Dual Code] - pdf with some examples and explanations

[[Category:编码理论]]