Dual code
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.
References
- ^ Conway, J.H.; Sloane,N.J.A. (1988). Sphere packings, lattices and groups. Grundlehren der mathematischen Wissenschaften. 290. Springer-Verlag. p. 77. ISBN 0-387-96617-X.
- Hill, Raymond (1986). A first course in coding theory. Oxford Applied Mathematics and Computing Science Series. Oxford University Press. p. 67. ISBN 0-19-853803-0.
- Pless, Vera (1982). Introduction to the theory of error-correcting codes. Wiley-Interscience Series in Discrete Mathematics. John Wiley & Sons. p. 8. ISBN 0-471-08684-3.
- J.H. van Lint (1992). Introduction to Coding Theory. GTM. 86 (2nd ed ed.). Springer-Verlag. p. 34. ISBN 3-540-54894-7.