Dual code

In coding theory, the dual code of a linear code

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

is the linear code defined by

C ⊥ = { x ∈ F q n ∣ ⟨ x , c ⟩ = 0 ∀ c ∈ C } {\displaystyle C^{\perp }=\{x\in \mathbb {F} _{q}^{n}\mid \langle x,c\rangle =0\;\forall c\in C\}}

where

⟨ x , c ⟩ = ∑ i = 1 n x i c i {\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 ⟨ ⋅ ⟩ {\displaystyle \langle \cdot \rangle } . The dimension of C and its dual always add up to the length n:

dim ⁡ C + dim ⁡ C ⊥ = n . {\displaystyle \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 {\displaystyle 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 ] {\displaystyle G=[I_{k}|A]} , then the dual code C ⊥ {\displaystyle C^{\perp }} has generator matrix [ − A ¯ T | I k ] {\displaystyle [-{\bar {A}}^{T}|I_{k}]} , where I k {\displaystyle I_{k}} is the ( n / 2 ) × ( n / 2 ) {\displaystyle (n/2)\times (n/2)} identity matrix and a ¯ = a q ∈ F q {\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. Vol. 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. Vol. 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