Dual code: Difference between revisions

Revision as of 08:08, 11 July 2008

Revision as of 11:02, 11 July 2008

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and ''p'' is the [[characteristic (algebra)|characteristic]] of '''F'''_''q''. In [[linear algebra]] terms, the dual code is the [[annihilator]] of ''C'' with respect to the [[bilinear form]] <,>. The [[Dimension_(vector_space)|dimension]] of ''C'' and its dual always add up to ''n'':

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

The dual of the dual code is always the original code.

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.

For binary codes, the dual code consists of all code words, as [[binary string]]s, that have 1s in places overlapping the 1s in each word from ''C'' always at an [[even number]] of locations.

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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. Self-dual codes can be classified into four types<ref>{{cite book | last=Conway | first=J.H. | authorlink=John 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 | pages=77}}</ref>:

*'''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]]).

*'''Type II''' codes are binary self-dual codes which are doubly-even.

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==References==

{{reflist}}

{{unreferenced|date=July 2008}}

{{refbegin}}

* {{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 | pages=34}}

{{refend}}

[[Category:Coding theory]]