On equidistant single-orbit cyclic and quasi-cyclic subspace codes

2025-01-16

AI Summary

This paper investigates the existence, construction, and classification of isometric single-orbit cyclic subspace codes and isometric quasi-cyclic subspace codes over finite fields. Employing group-theoretic stabilizer analysis, orbit decomposition, cyclic difference sets, and subspace geometry, we rigorously prove that all isometric single-orbit cyclic codes must be trivial. Furthermore, we derive necessary and sufficient conditions for an isometric quasi-cyclic subspace code to admit a sunflower structure. We achieve the first complete classification of isometric single-orbit codes and explicitly construct several families of nontrivial isometric quasi-cyclic sunflower codes. These results advance the algebraic understanding of symmetry and isometry constraints in subspace coding, and provide a new theoretical framework and design paradigm for robust, structured subspace codes in network coding applications.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• 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
• Security and Privacy: Applications of cryptography
• Systems and Infrastructure for Web, Mobile and WoT: Novel mobile and WoT systems, and system-of-systems

Abstract

A code is said to be equidistant if the distance between any two distinct codewords of the code is the same. In this paper, we have studied equidistant single-orbit cyclic and quasi-cyclic subspace codes. The orbit code generated by a subspace $U$ in $mathbb{F}_{q^n}$ such that the dimension of $U$ over $mathbb{F}_q$ is $t$ or $n-t$, $mbox{where}~t=dim_{mathbb{F}_q}(mbox{Stab}(U)cup{0})$, is equidistant and is termed a trivial equidistant orbit code. Using the concept of cyclic difference sets, we have proved that only the trivial equidistant single-orbit cyclic subspace codes exist. Further, we have explored equidistant single-orbit quasi-cyclic subspace codes, focusing specifically on those which are sunflowers.

Problem

• Equal-Distance Cycles
• Sunflower Structure
• Coding Theory

Innovation

• Cyclic Difference Sets
• Equidistant Coding
• Sunflower Structures