Bounding the Trellis State Complexity of Algebraic Geometric Codes

Abstract

Let ** be an algebraic geometric code of dimension k and length n constructed on a curve ** over Fq. Let ** be the state complexity of ** and ** the Wolf upper bound on **. We introduce a numerical function R that depends on the gonality sequence of ** and show that ** where g is the genus of **. As a matter of fact, R(2g−2)≤g−(γ2−2) with γ2 being the gonality of ** over Fq, and thus in particular we have that **.