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| {{for|players of both rugby codes|List of dual-code rugby internationals}}
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| In [[coding theory]], the '''dual code''' of a [[linear code]]
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| :<math>C\subset\mathbb{F}_q^n</math> | |
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| is the linear code defined by
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| :<math>C^\perp = \{x \in \mathbb{F}_q^n \mid \langle x,c\rangle = 0\;\forall c \in C \} </math>
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| where
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| :<math>\langle x, c \rangle = \sum_{i=1}^n x_i c_i </math>
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| is a scalar product. In [[linear algebra]] terms, the dual code is the [[Annihilator_(ring_theory)|annihilator]] of ''C'' with respect to the [[bilinear form]] <,>. The [[Dimension_(vector_space)|dimension]] of ''C'' and its dual always add up to the length ''n'':
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| :<math>\dim C + \dim C^\perp = n.</math>
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| A [[generator matrix]] for the dual code is a [[parity-check matrix]] for the original code and vice versa. The dual of the dual code is always the original code.
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| ==Self-dual codes== | |
| A '''self-dual code''' is one which is its own dual. This implies that ''n'' is even and dim ''C'' = ''n''/2. If a self-dual code is such that each codeword's weight is a multiple of some constant <math>c > 1</math>, then it is of one of the following four types:<ref>{{cite book | last=Conway | first=J.H. | authorlink=John Horton Conway | coauthors=Sloane,N.J.A. | authorlink2=Neil Sloane | title=Sphere packings, lattices and groups | series=Grundlehren der mathematischen Wissenschaften | volume=290 | publisher=[[Springer-Verlag]] | date=1988 | isbn=0-387-96617-X | page=77}}</ref>
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| *'''Type I''' codes are binary self-dual codes which are not [[doubly even code|doubly even]]. Type I codes are always [[even code|even]] (every codeword has even [[Hamming weight]]).
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| *'''Type II''' codes are binary self-dual codes which are doubly even.
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| *'''Type III''' codes are ternary self-dual codes. Every codeword in a Type III code has Hamming weight divisible by 3.
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| *'''Type IV''' codes are self-dual codes over '''F'''<sub>4</sub>. These are again even.
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| Codes of types I, II, III, or IV exist only if the length ''n'' is a multiple of 2, 8, 4, or 2 respectively.
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| ==References==
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| {{reflist}}
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| {{refbegin}}
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| * {{cite book | last=Hill | first=Raymond | title=A first course in coding theory | publisher=[[Oxford University Press]] | series=Oxford Applied Mathematics and Computing Science Series | date=1986 | isbn=0-19-853803-0 | page=67 }}
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| * {{cite book | last = Pless | first = Vera | authorlink=Vera Pless | title = Introduction to the theory of error-correcting codes | publisher = [[John Wiley & Sons]]|series = Wiley-Interscience Series in Discrete Mathematics | date = 1982| isbn = 0-471-08684-3 | page=8 }}
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| * {{cite book | author=J.H. van Lint | title=Introduction to Coding Theory | edition=2nd ed | publisher=Springer-Verlag | series=[[Graduate Texts in Mathematics|GTM]] | volume=86 | date=1992 | isbn=3-540-54894-7 | page=34}}
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| {{refend}}
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| == External links ==
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| * [http://www.maths.manchester.ac.uk/~pas/code/notes/part9.pdf MATH32031: Coding Theory - Dual Code] - pdf with some examples and explanations
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| [[Category:Coding theory]]
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