is the linear code defined by
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:
- 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 .
- 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.). Springer-Verlag. p. 34. ISBN 3-540-54894-7.
- MATH32031: Coding Theory - Dual Code - pdf with some examples and explanations