Introduction to Cryptography with Coding Theory, 3rd Edition

24.7 Cyclic Codes

Cyclic codes are a very important class of codes. In the next two sections, we’ll meet two of the most useful examples of these codes. In this section, we describe the general framework.

A code $C$ is called cyclic if

$$(c_1\text{,}{c}_2\text{,}\dots \text{,}{c}_n)\in C\text{ implies }(c_n\text{,}{c}_1\text{,}{c}_2\text{,}\dots \text{,}{c}_{n-1})\in C\text{.}$$

For example, if $(1\text{,}1\text{,}0\text{,}1)$ is in a cyclic code, then so is $(1\text{,}1\text{,}1\text{,}0)$. Applying the definition two more times, we see that $(0\text{,}1\text{,}1\text{,}1)$ and $(1\text{,}0\text{,}1\text{,}1)$ are also codewords, so all cyclic permutations of the codeword are codewords. This might seem to be a strange condition for a code to satisfy. After all, it would seem to be rather irrelevant that, for a given codeword, all of its cyclic shifts are still codewords. The point is that cyclic codes have a lot of ...