Høholdt, van Lint, and Pellikaan used order functions to construct codes by means of Linear Algebra and Semigroup Theory only. However, Geometric Goppa codes that can be represented by this method are mainly those based on just one point. In this correspondence, we introduce the concept of near order function with the aim of generalizing this approach in such a way that a wider family of Geometric Goppa codes can be studied on a more elementary setting.

For a linear code C of length n and dimension k, Wolf (1978) noticed that the trellis state complexity s(C) of C is upper-bounded by w(C):=min(k,n-k). We point out some new lower bounds for s(C). In particular, if C is an algebraic-geometric code, then s(C)/spl ges/w(C)-(g-a), where g is the genus of the underlying curve and a is the abundance of the code.

Improved geometric Goppa codes have a smaller redundancy and the same bound on the minimum distance as ordinary algebraic-geometry codes. For an asymptotically good sequence of function fields we give a formula for the redundancy.