“Dual code”的意思、由来-开放百科全书

词条

Dual code

释义

Self-dual codes
References
External links

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

In coding theory, the dual code of a linear code

is the linear code defined by

where

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:

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 , 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.

If a self-dual code has a generator matrix of the form , then the dual code has generator matrix , where is the identity matrix and .

References

1. ^{{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=77}}

{{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 | publisher=Springer-Verlag | series=GTM | volume=86 | date=1992 | isbn=3-540-54894-7 | page=34}}

External links

MATH32031: Coding Theory - Dual Code - pdf with some examples and explanations

1 : Coding theory

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