Dual code - Alchetron, The Free Social Encyclopedia

In coding theory, the dual code of a linear code

C ⊂ F q n

is the linear code defined by

C ⊥ = { x ∈ F q n ∣ ⟨ x , c ⟩ = 0 ∀ c ∈ C }

where

⟨ x , c ⟩ = ∑ 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:

dim ⁡ C + dim ⁡ C ⊥ = 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:

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 ⊥ has generator matrix [ − A ¯ T | I k ] , where I k is the ( n / 2 ) × ( n / 2 ) identity matrix and a ¯ = a q ∈ F q .

References

Dual code Wikipedia

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