Latroids and code invariants

Abstract

Latroids were introduced by Vertigan, who associated a latroid to a linear block code and showed that its Tutte polynomial determines the weight enumerator of the code. The original definition of a latroid is in terms of its rank function. For a complemented lattice, we establish cryptomorphic definitions in terms of independent elements, bases, circuits, and flats. We then associate a latroid to a code over a ring or a field endowed with a general support function and show that the generalized weights of the code can be recovered from the associated latroid. This provides a uniform framework for studying generalized weights and other combinatorial invariants of linear block codes, linear codes over a ring, rank-metric, and sum-rank metric codes.