PERSPECTIVES

    I use Gauss sums over finite semi-simple algebras to give the divisibility and the weight of abelian codes [PL, 1998].

    there exists examples of  constructions of sequences from the Galois rings$GR(4,f)$[ Sole, Boztas, Kumar, 1994]

We may hope new results replacing the finite fields by finite rings.

    They are complete Gauss sums

$$G_A(\chi,\psi) =\sum_{x\in A^\times} \chi(x)\psi(x)$$

    and incomplete Gauss sums

$$G_X(\chi,\psi) =\sum_{x\in X\cap A^\times} \chi(x)\psi(x)$$

where $X$ is a subgroup of $A^\times$. For applications in designs theory there is no restriction on $X$ but for sequences point of view $X$ must be cyclic. Namely, if$A$ is local there are Techmüller Gauss sums.

    The evaluation of Gauss sums is not easy on finite field and harder on finite rings but there are good news :

   Local Frobenius ring : the modulus of complete Gauss is easy calculate [PL, 1998].

   Galois rings : The Teichmüller Gauss sums are well understand in terms of curves [KHC, 1996].

    Galois rings of characteristic have like Stickelberger's congruences properties, [PL, 1998].

    Ramified rings : good family of sequences can be construct from such rings, [PL, 1998]. for example :

$$B = GR(4, f)[X]/(X^2-2, 2X)$$