Dual code

In coding theory, the dual code of a linear code

${\displaystyle C\subset \mathbb {F} _{q}^{n}}$

is the linear code defined by

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

where

${\displaystyle \langle x,c\rangle =\sum _{i=1}^{n}x_{i}c_{i}}$

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

${\displaystyle \dim C+\dim C^{\perp }=n.}$

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 $c > 1$, 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 ${\displaystyle G=[I_{k}|A]}$, then the dual code ${\displaystyle C^{\perp }}$ has generator matrix ${\displaystyle [-{\bar {A}}^{T}|I_{k}]}$, where $I_{k}$ is the ${\displaystyle (n/2)\times (n/2)}$ identity matrix and ${\displaystyle {\bar {a}}=a^{q}\in \mathbb {F} _{q}}$.

References

1. 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.). Springer-Verlag. p. 34. ISBN 3-540-54894-7.

External links

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