From weight enumerators to zeta functions

In [2], we introduced, for an arbitrary linear code, its zeta function. The definition is motivated by properties of algebraic curves and of codes constructed with these curves. In this paper, we give an alternative but equivalent definition in terms of the puncturing and shortening operators acting on a linear code. For certain infinite families of divisible codes, we compute the zeta functions. With the notion of a zeta function, an analogue of the Riemann hypothesis can be formulated for codes. We show the relation between such a Riemann hypothesis and upper bounds on the parameters of linear codes. The proof of the Riemann hypothesis analogue is open and the upper bounds are conjectural.