On the Riemann hypothesis for self-dual weight enumerators of genera three and four

Abstract

. Zeta functions for linear codes were defined by I. Duursma in 1999. In the cases of genera less than three, S. Nishimura gave equivalent conditions for their Riemann hypothesis. In this paper, using a new method, we give similar equivalent conditions for the cases of genera three and four. Our method can be applied to smaller genera and leads to an alternative simple proofs of Nishimura’s theorems. Using these results, we examine the Riemann hypothesis of some invariant polynomials. We also discuss the cases of genera greater than four and propose some new problems.