Harder–Narasimhan theory for linear codes (with an appendix on Riemann–Roch theory)

Abstract

In this text we develop some aspects of Harder–Narasimhan theory, slopes, semistability and canonical filtration, in the setting of combinatorial lattices. Of noticeable importance is the Harder–Narasimhan structure associated to a Galois connection between two lattices. It applies, in particular, to matroids.We then specialize this to linear codes. This could be done from at least three different approaches: using the sphere-packing analogy, or the geometric view, or the Galois connection construction just introduced. A remarkable fact is that these all lead to the same notion of semistability and canonical filtration. Relations to previous propositions toward a classification of codes, and to Wei's generalized Hamming weight hierarchy, are also discussed.Last, we study the two important questions of the preservation of semistability (or more generally the behavior of slopes) under duality, and under tensor product. The former essentially follows from Wei's duality theorem for higher weights—and its matroid version—which we revisit in an appendix, developing analogues of the Riemann–Roch, Serre duality, Clifford, and gap and gonality sequence theorems. Likewise the latter is closely related to the bound on higher weights of a tensor product, conjectured by Wei and Yang, and proved by Schaathun in the geometric language, which we reformulate directly in terms of codes. From this material we then derive semistability of tensor product.