Cyclic code

In coding theory, a cyclic code is a block code, where the circular shifts of each codeword gives another word that belongs to the code. They are error-correcting codes that have algebraic properties that are convenient for efficient error detection and correction.

Let ${\displaystyle {\mathcal {C}}}$ be a linear code over a finite field (also called Galois field) ${\displaystyle GF(q)}$ of block length n. ${\displaystyle {\mathcal {C}}}$ is called a cyclic code if, for every codeword c=(c₁,...,c_n) from C, the word (c_n,c₁,...,c_n-1) in ${\displaystyle GF(q)^{n}}$ obtained by a cyclic right shift of components is again a codeword. Because one cyclic right shift is equal to n − 1 cyclic left shifts, a cyclic code may also be defined via cyclic left shifts. Therefore the linear code ${\displaystyle {\mathcal {C}}}$ is cyclic precisely when it is invariant under all cyclic shifts.

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