Latroids and code invariants

2025-03-04

AI Summary

Unifying the characterization of generalized weights and weight distributions across diverse code families—including linear block codes, linear codes over rings, rank-metric codes, and sum-rank metric codes—remains an open challenge, as existing theories are largely restricted to linear codes over fields. Method: We introduce the *latroid*, a lattice-theoretic structure defined via a generalized support function, thereby extending matroid theory to arbitrary support functions. We construct natural mappings from each code family to latroids and employ combinatorial geometry, Tutte polynomials, and support-function analysis. Contribution/Results: We rigorously prove that the latroid fully determines the generalized weight hierarchy; moreover, under a weak separability condition on the support function, the complete weight distribution is reconstructible. This framework transcends classical field-based limitations and establishes the first unified, algebraic, and combinatorial invariant theory applicable to multiple non-standard code families.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Probabilistic Circuits and Graphical Models
• Game Theory and Economic Paradigms: Cooperative Game Theory

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

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.

Problem

• Associates latroids with codes over rings or fields
• Recovers generalized weights from associated latroids
• Provides framework for studying various code types

Innovation

• Latroids linked to codes with support functions
• Generalized weights derived from associated latroids
• Latroids determine code weight distribution