Latroids and code invariants
Elisa Gorla, Flavio Salizzoni
TL;DR
The paper develops a unified lattice-theoretic framework—latroids—for studying invariants of linear codes over rings and in various metrics. By constructing latroids from codes via modular supports and chain supports, it shows how generalized weights and, under suitable conditions, weight distributions can be recovered from the latroid and its Tutte-Whitney polynomial. It bridges block codes, ring-linear codes, rank-metric codes, and sum-rank codes, connecting generalized weights to q-matroid/q-polymatroid and sum-matroid theories within a single combinatorial object. This provides a versatile toolkit for deriving invariants, understanding equivalence, and translating between matroid-like theories and coding-theoretic properties with potential computational benefits.
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. We 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 of linear block codes, linear codes over a ring, rank-metric and sum-rank metric codes. Under suitable assumptions, we show that the latroid determines the weight distribution of the code.