Harder-Narasimhan theory for linear codes

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 they all lead to the same notion of semistability and canonical filtration. Relations to previous propositions towards a classification of codes, and to Wei’s generalized Hamming weight hierarchy, are also discussed. Last, we study the important question of the preservation of semistability (or more generally the behaviour of slopes) under duality, and under tensor product. The former essentially follows from Wei’s duality theorem for higher weights, which we revisit in developing analogues of the Riemann-Roch, Serre duality, and gap theorems for codes. The latter is shown likewise to follow from 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.