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# linear code

Often in coding 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 *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.

## Mathematics Subject Classification

94B05*no label found*

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