Dual code: Difference between revisions

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CommunicationIn Theory.[[communication theory]], Thethe '''dual code''' of a [[linear code]] <math>C\subset\mathbb{F}_2^n</math> is defined to be

:<math>C^\perp = \{x \in subset\mathbb{F}_2^n \mid <x,c> = 0 \forall c \in C \} </math>

is the linear code defined by

where <,> denotes the vector [[dot product]] (which is taken over the [[field (mathematics)|field]] <math>\mathbb{F}_2</math>).

:<math>C^\perp = \{x \in \mathbb{F}_2^n \mid <x,c> = 0 \forall c \in C \} </math>

An important property is that the dual of the dual code is the original code itself.

Here <,> denotes the vector [[dot product]], which is taken over the [[field (mathematics)|field]] <math>\mathbb{F}_2</math>. In simpler language, it consists of all code words, as [[binary string]]s, that have 1s in places overlapping the 1s in each word from ''C'' always at an [[even number]] of locations.

In [[linear algebra]] terms, the dual code is the [[annihilator]] of ''C'' with respect to the [[bilinear form]] <,>. An important property is that the dual of the dual code is the original code itself. This follows from the fact that the dimensions of ''C'' and its dual always add up to ''n''.

[[Category:Coding theory]]

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