linear code

Often in coding , a code’s alphabet is taken to be a finite field. In particular, if A is the finite field with two (resp. three, four, etc.) elements, we call C a binary (resp. ternary, quaternary, etc.) code. In particular, when our alphabet is a finite field then the set An is a vector spaceMathworldPlanetmath over A, and we define a linear codeMathworldPlanetmath over A of block length n to be a subspacePlanetmathPlanetmathPlanetmath (as opposed to merely a subset) of An. We define the dimensionPlanetmathPlanetmath of C to be its dimension as a vector space over A.

Though not sufficient for unique classification, a linear code’s block length, dimension, and minimum distance are three crucial parameters in determining the strength of the code. For referencing, a linear code with block length n, dimension k, and minimum distance d is referred to as an (n,k,d)-code.

Some examples of linear codes are Hamming Codes, BCH codes, Goppa codes, Reed-Solomon codes, and the Golay code (

Title linear code
Canonical name LinearCode
Date of creation 2013-03-22 14:21:24
Last modified on 2013-03-22 14:21:24
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 7
Author mathcam (2727)
Entry type Definition
Classification msc 94B05
Related topic CyclicCode
Related topic WeightEnumerator
Related topic DualCode
Related topic EvenCode
Related topic AutomorphismGroupLinearCode
Defines binary code
Defines ternary code
Defines quaternary code
Defines dimension of a linear code