Dual code

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title: "Dual code" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["coding-theory"] topic_path: "general/coding-theory" source: "https://en.wikipedia.org/wiki/Dual_code" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

In coding theory, the dual code of a linear code

:C\subset\mathbb{F}_q^n

is the linear code defined by

:C^\perp = {x \in \mathbb{F}_q^n \mid \langle x,c\rangle = 0;\forall c \in C }

where

:\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 \langle\cdot\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:

  • 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)\times (n/2) identity matrix and \bar{a}=a^q\in\mathbb{F}_q.

References

References

  1. Conway, J.H.. (1988). "Sphere packings, lattices and groups". [[Springer-Verlag]].

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