Dual code: Difference between revisions

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{{for|players of both rugby codes see|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]] <,math>\langle\cdot\rangle</math>. 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 athe [[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 | coauthorsauthor2=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 code|doubly even]]. Type I codes are always [[even code|even]] (every codeword has even [[Hamming weight]]).

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 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 III''' 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'''₄. 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 ed | publisher=Springer-Verlag | series=[[Graduate Texts in Mathematics|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 ==

[[Category:Coding theory]]

* [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

[[csCategory:DuálníCoding kódtheory]]