Codes over finite quotients of polynomial rings

Thierry Pierre Berger (1) , Nora El Amrani (1)

1 XLIM-DMI - DMI

In this paper, we study codes that are defined over the polynomial ring A=F[x]/f(x), where f(x) is a monic polynomial over a finite field F. We are interested in codes that are A-submodules of Aℓ. These codes are a generalization of quasi-cyclic codes. In this work we introduce a notion of basis of divisors for these codes and a canonical generator matrix. It is a generalization of the work of K. Lally and P. Fitzpatrick. However, in contrast with K. Lally and P. Fitzpatrick, we do not use the Gröbner basis, but only the classical Euclidean division. We also study the notion of A-duality and the link with the q-ary images of these codes and the F-duality.

Mots clés: Cyclic codes; Quasi-cyclic codes; Polynomial ring; q-ary image; Duality

Thierry Pierre Berger, Nora El Amrani. Codes over finite quotients of polynomial rings. Finite Fields and Their Applications, 2014, 25, pp.165-181. ⟨10.1016/j.ffa.2013.09.004⟩. ⟨hal-00913825⟩