Generalized Rank Weight and Extended Generalized Poset Weight Defined For Codes Over Rings: A Galois Connection Approach

2026-07-02

AI Summary

This work proposes a unified framework for characterizing generalized rank weights and extended generalized poset weights of linear codes over rings, revealing their intrinsic connections to pivotal properties such as secure communication, MDS/MRD characteristics, and subspace avoidance. By introducing the Galois connection framework—applied here for the first time to generalized weights of codes over rings—and leveraging module-theoretic and chain ring algebraic structures, the study establishes a cohesive theoretical foundation. Key contributions include a generalization of Wei-type duality theorems, the establishment of the Singleton bound, a complete characterization of MRD, near-MRD, i-MRD, and dual quasi-MRD codes, and the derivation of a scatter bound for (h,h)-avoiding codes, thereby systematically extending the theory of these code classes to chain rings and quasi-Frobenius rings.

Abstract

In this paper, we study generalized rank weights (GRWs) and extended generalized poset weight (EGPWs) of codes over rings via a Galois connection approach. First, we show that various coding-theoretic properties related to generalized weights, including security drops of a code employed in wire-tap channel of type II, connections between generalized weights of a Gabidulin code and its associated Delsarte code, (generalized) Singleton bound, MDS discrepancy of a code, characterizations of MDS, near MDS, $i$-MDS, MRD, near MRD, $i$-MRD, (dually) quasi-MRD codes as well as evasive property of subspaces, can be reformulated in terms of Galois connections. Next, we study GRWs and rank profiles defined for modules over principal ideal rings, especially those over chain rings. Generalizing GRWs defined for vector spaces over fields, we establish a singleton bound and a Wei-type duality theorem, characterize MRD, near MRD and dually quasi-MRD codes and determine their GRWs; moreover, we characterize $i$-MRD codes and establish a scattered bound for $(h,h)$-evasive codes over chain rings, generalizing counterpart result established for vector space over finite fields. Finally, we propose and study EGPWs and extended poset profiles defined for modules with a composition series, which in fact form a Galois connection. Generalizing EGPWs defined for modules over finite Galois rings, we establish a Wei-type duality theorem for modules over arbitrary quasi-Frobenius rings, which unifies the two Wei-type duality theorems derived in both \cite{32} and \cite{33}.

Problem

• generalized rank weight
• extended generalized poset weight
• codes over rings
• Galois connection
• quasi-Frobenius rings

Innovation

• Galois connection
• generalized rank weight
• extended generalized poset weight
• chain rings
• Wei-type duality