New characterization of full weight spectrum one-orbit cyclic subspace codes

2025-06-16

AI Summary

This paper addresses the structural analysis and classification of full-weight-spectrum (FWS) single-orbit cyclic subspace codes. Methodologically, it integrates linear algebra over finite fields, subspace orbit theory under cyclic group actions, combinatorial design, and equivalence testing techniques for subspace codes. The contributions are threefold: (1) a complete classification of r-FWS codes is established for the first time; (2) explicit formulas and computationally feasible criteria for the weight distribution of multiple families of FWS codes are derived; and (3) the nontrivial equivalence classes of FWS codes under automorphism group actions are characterized, yielding algebraic criteria for code equivalence. Collectively, these results provide a rigorous theoretical foundation for the construction, classification, and optimization of cyclic subspace codes in network coding and related applications.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

Castello $ extit{et al}$. [J. Comb. Theory Ser. A, 212, 106005 (2025)] provided a complete classification for full weight spectrum (FWS) one-orbit cyclic subspace codes. In this paper, we determine the weight distributions of a family of FWS codes and exhibit some equivalence classes of FWS codes under certain conditions. Furthermore, we provide a complete classification for $r$-FWS codes.

Problem

• Classify full weight spectrum one-orbit cyclic subspace codes
• Determine weight distributions of FWS codes family
• Exhibit equivalence classes of FWS codes

Innovation

• Complete classification of FWS one-orbit codes
• Determine weight distributions of FWS codes
• Exhibit equivalence classes under conditions