Multilevel inserting constructions for constant dimension subspace codes

2025-08-05

AI Summary

This paper addresses the limited codebook size in constructing constant-dimension codes (CDCs). To overcome this limitation, we propose the *inverse bilaterally multilevel construction*, which unifies the dual-multilevel and bilaterally multilevel frameworks. Our approach introduces novel concepts: inverse bilaterally indexed vectors and inverse bilaterally Ferrers diagram rank-metric codes, along with a new multilevel insertion structure and an optimized set of bilaterally indexed vectors. The proposed method significantly improves the lower bound on CDC sizes: for $q geq 3$, the new lower bound achieves at least $0.94548$ of the best known upper bound—substantially closing the gap to the theoretical limit. Moreover, several constructed CDCs exceed all previously reported sizes in the literature. These advances provide superior constant-dimension rank-metric codes for efficient error correction in random network coding.

Technology Category

• Machine Learning: Large Multimodal Models (LMMs)
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

Subspace codes, especially constant dimension subspace codes (CDCs), represent an intriguing domain that can be used to conduct basic coding theory investigations. They have received widespread attention due to their applications in random network coding. This paper presents inverse bilateral multilevel construction by introducing inverse bilateral identifying vectors and inverse bilateral Ferrers diagram rank-metric codes. By inserting the inverse bilateral multilevel construction into the double multilevel construction and bilateral multilevel construction, an effective construction for CDCs is provided. Furthermore, via providing a new set of bilateral identifying vectors, we give another efficient construction for CDCs. In this article, several CDCs are exhibited, equipped with the rank-metric, with larger sizes than the known ones in the existing literature. From a practical standpoint, our results could help in the pragmatic framework of constant-dimension-lifted rank-metric codes for applications in network coding. The ratio of the new lower bound to the known upper bound for some CDCs is calculated, which is greater than 0.94548 for any prime power $q geq 3.$

Problem

• Constructs larger constant dimension subspace codes (CDCs)
• Improves CDCs via inverse bilateral multilevel techniques)
• Enhances network coding applications with new lower bounds)

Innovation

• Inverse bilateral multilevel construction for CDCs
• New bilateral identifying vectors set
• Larger CDC sizes with rank-metric