Dual code: Difference between revisions

Revision as of 19:02, 7 May 2014

Revision as of 09:53, 5 August 2015

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{{for|players of both rugby codes|List of dual-code rugby internationals}}

In [[coding theory]], the '''dual code''' of a [[linear code]]

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

is the linear code defined by

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:<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_Annihilator (ring_theoryring theory)|annihilator]] of ''C'' with respect to the [[bilinear form]] <,>. The [[Dimension_Dimension (vector_spacevector space)|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 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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* {{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 }}

* {{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 }}

* {{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}}

{{refend}}