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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_(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'':
 
:<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.
 
==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>
*'''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.
*'''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.
 
==References==
{{reflist}}
{{refbegin}}
* {{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}}
 
== External links ==
* [http://www.maths.manchester.ac.uk/~pas/code/notes/part9.pdf MATH32031: Coding Theory - Dual Code] - pdf with some examples and explanations
 
[[Category:Coding theory]]

Revision as of 20:55, 3 May 2013

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In coding theory, the dual code of a linear code

C⊂𝔽qn

is the linear code defined by

C⊥={x∈𝔽qn∣⟨x,c⟩=0∀c∈C}

where

⟨x,c⟩=∑i=1nxici

is a scalar product. In linear algebra terms, the dual code is the annihilator of C with respect to the bilinear form <,>. The dimension of C and its dual always add up to the length n:

dim⁡C+dim⁡C⊥=n.

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.

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 c>1, then it is of one of the following four types:[1]

  • 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 F4. 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.

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro. Template:Refbegin

  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

Template:Refend

External links

  1. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
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