TABLE OF CONTENTS After the theoretical results of C. Shannon and the first linear code constructions (Hamming, Golay) American engineers were required to be able to obtain codes stable not only under addition (linear codes), but also stable under circular sliding (or shift). The codes obtained (cyclic codes) are linear codes with additional properties.
This new requirement led the mathematicians to exploit the structure of A 
, and, in particular, to study the ideal A. An ideal A is a non-empty part, stable under addition, and stable under multiplication by any element of A. It is a cyclic code of length n. Everywhere hereinafter n is odd.
The following results express the properties of a cyclic code.
Any code C, stable under addition and circular shift may be represented as an ideal A.
| Proof | The circular shift on the right represents the multiplication by X in A. The code C is thus stable under addition and multiplication by X. It is therefore stable under addition and multiplication by any polynomial: thus, it is an ideal A. Conversely, an ideal A is clearly a code stable under addition and circular shift. |
Any cyclic code has the form ( g( X)) (i.e. the set of multiples of g( X)), with g( X) dividing X n ? 1. More precisely, there is between the cyclic codes of length...
