linear code
Often in coding , a code’s alphabet is taken to be a finite field. In particular, if is the finite field with two (resp. three, four, etc.) elements, we call a binary (resp. ternary, quaternary, etc.) code. In particular, when our alphabet is a finite field then the set is a vector space over , and we define a linear code over of block length to be a subspace (as opposed to merely a subset) of . We define the dimension of to be its dimension as a vector space over .
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 , dimension , and minimum distance is referred to as an -code.
Some examples of linear codes are Hamming Codes, BCH codes, Goppa codes, Reed-Solomon codes, and the Golay code (http://planetmath.org/BinaryGolayCode).
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 |