Dual code explained

In coding theory, the dual code of a linear code

C\subset F ^{n} _{q}

is the linear code defined by

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

where

\langle x, c \rangle = \sum ^{n} _{i =1} x _{i} c _{i} \langle ⋅ \rangle . 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:^[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

G = [I _{k|A]} , then the dual code C ^{\perp} has generator matrix [ - \bar{A} ^{T|I} _{k]} , where I _{k} is the (n/ 2 ) x (n/ 2 ) identity matrix and \bar{a} = a ^{q\in F} _{q} .

References

• Book: Hill, Raymond . A first course in coding theory . registration . . Oxford Applied Mathematics and Computing Science Series . 1986 . 0-19-853803-0 . 67 .
• Book: Pless, Vera . Vera Pless

. Vera Pless . Introduction to the theory of error-correcting codes. Introduction to the Theory of Error-Correcting Codes . John Wiley & Sons. Wiley-Interscience Series in Discrete Mathematics . 1982. 0-471-08684-3 . 8 .

• Book: J.H. van Lint . Introduction to Coding Theory . 2nd . Springer-Verlag . . 86 . 1992 . 3-540-54894-7 . 34 .

External links

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

Notes and References

1. Book: Conway, J.H. . John Horton Conway

   . John Horton Conway . Sloane, N.J.A. . Neil Sloane . Sphere packings, lattices and groups . Grundlehren der mathematischen Wissenschaften . 290 . . 1988 . 0-387-96617-X . 77 .