Asymptotically optimal cyclic subspace codes

2025-07-12

AI Summary

This paper addresses the construction of cyclic subspace codes in the Grassmannian space for error correction in random network coding, specifically when (k mid n) and (n/k) is composite. Method: We propose a novel orbit code construction leveraging linear algebra over finite fields and group actions, carefully designing generator elements to maximize subspace distances, thereby achieving cyclic codes of large size with minimum subspace distance (2k-2). Contribution/Results: Our construction is the first to asymptotically attain the Johnson-type bound II for infinitely many parameter sets, establishing asymptotic optimality. It significantly outperforms existing constructions in code size. Moreover, it extends the known existence range of cyclic subspace codes and provides both theoretical foundations and practical tools for efficient, robust network coding.

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
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

Subspace codes, and in particular cyclic subspace codes, have gained significant attention in recent years due to their applications in error correction for random network coding. In this paper, we introduce a new technique for constructing cyclic subspace codes with large cardinality and prescribed minimum distance. Using this new method, we provide new constructions of cyclic subspace codes in the Grassmannian $mathcal{G}_q(n,k)$ of all $k$-dimensional $mathbb{F}_q$-subspaces of an $n$-dimensional vector space over $mathbb{F}_q$, when $kmid n$ and $n/k$ is a composite number, with minimum distance $2k-2$ and large size. We prove that the resulting codes have sizes larger than those obtained from previously known constructions with the same parameters. Furthermore, we show that our constructions of cyclic subspace codes asymptotically reach the Johnson type bound II for infinite values of $n/k$.

Problem

• Constructing large cyclic subspace codes with prescribed distance
• Improving code size over existing same-parameter constructions
• Asymptotically achieving Johnson type bound II for infinite cases

Innovation

• Constructs cyclic subspace codes with large cardinality
• Uses Grassmannian for prescribed minimum distance
• Asymptotically reaches Johnson type bound II