Homogeneous weights and exponential sums. (Q1400977)

Let \({\mathbb F}_{q}\) be a finite field of characteristic \(p\) with \(q=p^{ \mu}\) elements, and \(W_{l}({\mathbb F}_{q})\) the ring of Witt vectors of length \(l\) over \({\mathbb F}_{q}\). The ring \(W_{l}({\mathbb F}_{q})\) is a finite local ring with \(q^{l}\) elements. The maximal ideal of \(W_{l}({\mathbb F}_{q})\) is generated by \(p\), and \(W_{l}({\mathbb F}_{q})\) is isomorphic to the Galois ring \(R\) which is the degree \( \mu\) extension of \({\mathbb Z}/p^{l}{\mathbb Z} \backsimeq W_{l}({\mathbb F}_{p})\). Generalizing the Goppa construction of linear codes coming from a smooth projective curve defined over \({\mathbb F}_{q}\), the second author [J. Pure Appl. Algebra, 144, 91--110 (1999; Zbl 0949.94007)] has constructed linear codes coming from an algebraic curve defined over \(W_{l}({\mathbb F}_{q})\) (a smooth connected irreducible scheme over \(\operatorname{Spec} W_{l}({\mathbb F}_{q})\) of relative dimension one). In this paper, the authors find explicitly the Fourier expansion (a linear combination of additive characters of \(W_{l}({\mathbb F}_{q})\)) for the homogeneous weight \(w(x)\) of \(x \in W_{l}({ \mathbb F}_{q})\) introduced by \textit{I. Constantinescu} and \textit{W. Heise} [Probl. Inf. Transm. 33, 208--213 (1997; Zbl 0977.94055)] and then use the above mentioned explicit result to find a lower bound for the weight of geometric Goppa codes coming from algebraic curves defined over \(W_{l}({\mathbb F}_{q})\). Note that the above mentioned Fourier expansion of the function \(w(x)\) involves an exponential sum over \(W_{l}({\mathbb F}_{q})\) so that the problem on finding a lower bound for the weight of algebraic geometric codes under consideration is reduced to estimation of the related exponential sums.