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# cyclic code

Let $C$ be a linear code over a finite field $A$ of block length $n$. $C$ is called a *cyclic code*, if for every codeword $c=(c_{1},\ldots,c_{n})$ from $C$, the word $(c_{n},c_{1},\ldots,c_{{n-1}})\in A^{n}$ obtained by a cyclic right shift of components is also a codeword from $C$.

Sometimes, $C$ is called *the* $c$-cyclic code, if $C$ is the smallest cyclic code containing $c$, or, in other words, $C$ is the linear code generated by $c$ and all codewords obtained by cyclic shifts of its components.

For example, if $A=\mathbb{F}_{2}$ and $n=3$, the codewords contained in the $(1,1,0)$-cyclic code are precisely

$(0,0,0),(1,1,0),(0,1,1)\text{ and }(1,0,1).$ |

Related:

LinearCode, Code

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Definition

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Reference

## Mathematics Subject Classification

94B15*no label found*

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