w:Dual code

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In coding theory, the dual code of a linear code

: $C\subset\mathbb{F}_q^n$

is the linear code defined by

: $C^\perp = \{x \in \mathbb{F}_q^n \mid \langle x,c\rangle = 0\;\forall c \in C \}$

where

: $\langle x, c \rangle = \sum_{i=1}^n x_i c_i$

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

: $\dim C + \dim C^\perp = n.$

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 $c > 1$ , then it is of one of the following four types:{{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 }}

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.

If a self-dual code has a generator matrix of the form $G=[I_k|A]$ , then the dual code $C^\perp$ has generator matrix $[-\bar{A}^T|I_k]$ , where $I_k$ is the $(n/2)\times (n/2)$ identity matrix and $\bar{a}=a^q\in\mathbb{F}_q$ .

References

{{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 }}

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