Dual code

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

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