Open problems on cyclic codes ∗∗ Pascale Charpin ∗ Contents 1 Introduction 3 2 Different kinds of cyclic codes. 4 2.1 Notation .............................. 5 2.2 Definitions ............................. 6 2.3 Primitive and non primitive cyclic codes ............ 17 2.4 Affine-Invariant codes ....................... 21 2.4.1 The poset of affine-invariant codes ............ 22 2.4.2 Affine-invariant codes as ideals of A ........... 28 3 On parameters of cyclic codes. 34 3.1 Codewords and Newton identities ................ 35 3.2 Special locator polynomials ................... 47 3.3 On the minimum distance of BCH codes ............ 59 3.4 On the weight enumerators .................... 65 3.4.1 The Reed-Muller codes .................. 67 3.4.2 On cyclic codes with two zeros .............. 71 3.4.3 On irreducible cyclic codes ................ 83 3.5 Automorphism groups of cyclic codes .............. 88 3.6 Are all cyclic codes asymptotically bad ? ............ 90 * INRIA, Projet CODES, Domaine de Voluceau, Rocquencourt BP 105, 78153 Le Ches- nay Cedex, FRANCE. e-mail: Pascale.Charpin@inria.fr 1