# 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 $A^{n}$ is a vector space over $A$, and we define a linear code over $A$ of block length $n$ to be a subspace (as opposed to merely a subset) of $A^{n}$. We define the dimension 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 (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