# 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 $c=({c}_{1},\mathrm{\dots},{c}_{n})$ from $C$, the word $({c}_{n},{c}_{1},\mathrm{\dots},{c}_{n-1})\in {A}^{n}$ obtained by a right shift of 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 shifts of its .

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

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

Trivial examples of cyclic codes are ${A}^{n}$ itself and the code containing only the zero codeword.

Title | cyclic code |
---|---|

Canonical name | CyclicCode |

Date of creation | 2013-03-22 15:12:56 |

Last modified on | 2013-03-22 15:12:56 |

Owner | GrafZahl (9234) |

Last modified by | GrafZahl (9234) |

Numerical id | 6 |

Author | GrafZahl (9234) |

Entry type | Definition |

Classification | msc 94B15 |

Related topic | LinearCode |

Related topic | Code |