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 con structed with these curves; In this paper, we give an alternative but equiv alent definition in terms of the puncturing and shortening operators acting on a linear code. For certain infinite families of divisible codes, we com pute 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 b etween such a Riemann hypothesis and upper bounds on the parameters of line ar codes. The proof of the Riemann hypothesis analogue is open and the uppe r bounds are conjectural. (C) 2001 Elsevier Science B.V. All rights reserve d.