Linear code: Difference between revisions

Revision as of 16:10, 28 February 2007

Revision as of 16:22, 28 February 2007

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==Formal definition==

A '''linear code''' of length ''n'' and rank ''k'' is a [[linear subspace]] ''C'' with [[dimension (linear algebra)|dimension]] ''k'' of the [[vector space]] <math>{\mathbb{F}_q}^n</math> where <math>\mathbb{F}_q</math> is the [[finite field]] with ''q'' elements. Such a code with parameter ''q'' is called a ''q''-ary code (e.g., when ''q''&nbsp;=&nbsp;5, the code is a 5-ary code). Notable exceptions to this are the casesIf ''q''&nbsp;=&nbsp;2 andor ''q''&nbsp;=&nbsp;3, when the code is described as a binary andcode, or a ternary (respectively)code.

==Properties==