An introduction to linear and cyclic codes

Daniel Augot (1) , Emmanuela Orsini (2) , Emanuele Betti (3)

1 SECRET - Security, Cryptology and Transmissions
2 Department of Mathematics "Federigo Enriques"
3 Dipartimento di Matematica "Ulisse Dini"

Our purpose is to recall some basic aspects about linear and cyclic codes. We first briefly describe the role of error-correcting codes in communication. To do this we introduce, with examples, the concept of linear codes and their parameters, in particular the Hamming distance. A fundamental subclass of linear codes is given by cyclic codes, that enjoy a very interesting algebraic structure. In fact, cyclic codes can be viewed as ideals in a residue classes ring of univariate polynomials. BCH codes are the most studied family of cyclic codes, for which some efficient decoding algorithms are known, as the method of Sugiyama.

Mots clés: coding theory; cyclic codes

Domaines: Théorie de l'information [cs.IT]; Théorie de l'information et codage [math.IT]; Cryptographie et sécurité [cs.CR]

Daniel Augot, Emmanuela Orsini, Emanuele Betti. An introduction to linear and cyclic codes. Sala, Massimiliano and Mora, Teo and Perret, Ludovic and Sakata, Shojiro and Traverso, Carlo. Gröbner Bases, Coding, and Cryptography, Springer-Verlag, pp.47-68, 2009, Gröbner Bases, Coding, and Cryptography, ⟨10.1007/978-3-540-93806-4_4⟩. ⟨inria-00550061⟩