Linear code: Difference between revisions

Revision as of 18:55, 30 January 2022

Revision as of 02:50, 14 July 2022

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==Definition and parameters==

A '''linear code''' of length ''n'' and rankdimension ''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 is called a ''q''-ary code. If ''q''&nbsp;=&nbsp;2 or ''q''&nbsp;=&nbsp;3, the code is described as a '''binary code''', or a '''ternary code''' respectively. The vectors in ''C'' are called ''codewords''. The '''size''' of a code is the number of codewords and equals ''q''^''k''.

The '''weight''' of a codeword is the number of its elements that are nonzero and the '''distance''' between two codewords is the [[Hamming distance]] between them, that is, the number of elements in which they differ. The distance ''d'' of the linear code is the minimum weight of its nonzero codewords, or equivalently, the minimum distance between distinct codewords. A linear code of length ''n'', dimension ''k'', and distance ''d'' is called an [''n'',''k'',''d''] code.