Dual code

In coding theory, the dual code of a linear code

$C\subset {\mathbb{𝔽}}_q^n$

is the linear code defined by

$C^{\perp }=\{x\in {\mathbb{𝔽}}_q^n\mid ⟨x,c⟩=0\mathrm{\forall }c\in C\}$

where

$⟨x,c⟩=\sum _{i=1}^n{x}_i{c}_i$

is a scalar product. In linear algebra terms, the dual code is the annihilator of C with respect to the bilinear form $⟨\cdot ⟩$. 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:

• 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 $[-{\overline{A}}^T|I_k]$, where $I_k$ is the $(n/2)\times (n/2)$ identity matrix and $\overline{a}=a^q\in {\mathbb{𝔽}}_q$.

References

• 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. https://archive.org/details/firstcourseincod0000hill. 
• 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. https://archive.org/details/introductiontoco0000lint/page/34. 

External links

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