linear codeLinearCode2004-05-05 18:01:202004-06-08 17:16:04Definitionbinary codeternary codequaternary codedimension of a linear code% this is the default PlanetMath preamble. as your knowledge
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\renewcommand{\>}{\rangle}Often in coding \PMlinkescapetext{theory}, 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 \emph{linear code over $A$} of block length $n$ to be a subspace (as opposed to merely a subset) of $A^n$. We define the \emph{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 \PMlinkname{Golay code}{BinaryGolayCode}.