Linear code: Difference between revisions

Revision as of 06:35, 30 November 2018

Revision as of 07:00, 30 November 2018

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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 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 athe 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.

We want to give <math>\mathbb{F}_q^n</math> the standard basis because each coordinate represents a "bit" that is transmitted across a "noisy channel" with some small probability of transmission error (a [[binary symmetric channel]]). If some other basis is used then this model cannot be used and the Hamming metric does not measure the number of errors in transmission, as we want it to.