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In coding theory, the '''dual code''' of a linear code
:<math>C\subset\mathbb{F}_q^n</math>
is the linear code defined by
:<math>C^\perp = \{x \in \mathbb{F}_q^n \mid \langle x,c\rangle = 0\;\forall c \in C \} </math>
where
:<math>\langle x, c \rangle = \sum_{i=1}^n x_i c_i </math>
is a scalar product. In linear algebra terms, the dual code is the annihilator of ''C'' with respect to the bilinear form <math>\langle\cdot\rangle</math>. The dimension of ''C'' and its dual always add up to the length ''n'':
:<math>\dim C + \dim C^\perp = n.</math>
A generator matrix for the dual code is the parity-check matrix for the original code and vice versa. The dual of the dual code is always the original code.
==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 | author2=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=[https://archive.org/details/spherepackingsla0000conw/page/77 77] | url=https://archive.org/details/spherepackingsla0000conw/page/77 }}</ref> *'''Type I''' codes are binary self-dual codes which are not doubly even. Type I codes are always even (every codeword has even Hamming weight). *'''Type II''' codes are binary self-dual codes which are doubly even. *'''Type III''' codes are ternary self-dual codes. Every codeword in a Type III code has Hamming weight divisible by 3. *'''Type IV''' codes are self-dual codes over '''F'''<sub>4</sub>. These are again even.
Codes of types I, II, III, or IV exist only if the length ''n'' is a multiple of 2, 8, 4, or 2 respectively.
If a self-dual code has a generator matrix of the form <math>G=[I_k|A]</math>, then the dual code <math>C^\perp</math> has generator matrix <math>[-\bar{A}^T|I_k]</math>, where <math>I_k</math> is the <math>(n/2)\times (n/2)</math> identity matrix and <math>\bar{a}=a^q\in\mathbb{F}_q</math>.
==References== {{reflist}} {{refbegin}} * {{cite book | last=Hill | first=Raymond | title=A first course in coding theory | url=https://archive.org/details/firstcourseincod0000hill | url-access=registration | publisher=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|title-link= 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 }} * {{cite book | author=J.H. van Lint | title=Introduction to Coding Theory | edition=2nd | publisher=Springer-Verlag | series=GTM | volume=86 | date=1992 | isbn=3-540-54894-7 | page=[https://archive.org/details/introductiontoco0000lint/page/34 34] | url=https://archive.org/details/introductiontoco0000lint/page/34 }} {{refend}}
== External links == * [https://web.archive.org/web/20170516194659/https://www.maths.manchester.ac.uk/~pas/code/notes/part9.pdf MATH32031: Coding Theory - Dual Code] - pdf with some examples and explanations
Category:Coding theory