Classification of Three-Dimensional One-Orbit Cyclic Subspace Codes in Fqn

핵심 개념

This paper provides a classification of three-dimensional one-orbit cyclic subspace codes in the Grassmannian space Gq(n,3) over the finite field Fqn.

초록

The paper focuses on the study of three-dimensional one-orbit cyclic subspace codes in the Grassmannian space Gq(n,3) over the finite field Fqn. The authors consider three families of such codes:

The first family contains only the code Orb(Fq3), which has minimum distance 6.

The second family contains the optimum-distance codes, i.e., codes with minimum distance 4. These codes correspond to Sidon spaces.

The third family contains codes with minimum distance 2.

The authors introduce new invariants, such as the dimension of the square-span of a subspace and the span of a subspace over a larger field, to distinguish inequivalent classes of codes. They provide a classification result based on the dimension of the square-span of a representative of the code and study the equivalence problem for the codes in the third family.
For the codes in the second family, the authors investigate the equivalence problem under the assumption that a representative is contained in the sum of two multiplicative cosets of Fq3. They show that these codes can be represented using linearized polynomials and provide a characterization of the Sidon spaces in this case.

통계

None.

인용구

None.

핵심 통찰 요약

On one-orbit cyclic subspace codes of $\mathcal{G}_q(n,3)$

by Chiara Caste... 게시일 arxiv.org 05-06-2024

https://arxiv.org/pdf/2405.01652.pdf

더 깊은 질문

Can the classification results be extended to one-orbit cyclic subspace codes of higher dimensions

The classification results provided in the paper for three-dimensional one-orbit cyclic subspace codes with specific properties can potentially be extended to higher dimensions. However, the complexity and intricacies of higher-dimensional spaces may introduce additional challenges in the classification process. Extending the classification results to higher dimensions would require a thorough analysis of the properties and invariants specific to those dimensions. It may involve exploring new techniques and considerations to effectively classify and distinguish inequivalent classes of cyclic subspace codes in higher dimensions.

Are there other invariants that can be used to distinguish inequivalent classes of cyclic subspace codes

In addition to the invariants mentioned in the paper, such as the dimension of the square-span of a subspace and the span of a subspace over a larger field, there are other invariants that can be utilized to distinguish inequivalent classes of cyclic subspace codes. Some potential invariants that could be considered include the rank of the subspace, the intersection properties with specific subspaces, the structure of the generating polynomials, and the behavior of the subspace under certain transformations or automorphisms. By incorporating these additional invariants into the classification process, a more comprehensive and detailed classification of cyclic subspace codes can be achieved.

What are the connections between the properties of Sidon spaces and the optimum-distance cyclic subspace codes studied in this paper

The properties of Sidon spaces and the optimum-distance cyclic subspace codes studied in the paper are closely connected. Sidon spaces play a crucial role in the study of cyclic subspace codes with optimum distance, as they provide a framework for understanding the structure and properties of these codes. The existence of Sidon spaces is essential for the construction and analysis of optimum-distance codes, as they exhibit specific properties that are conducive to achieving the desired minimum distance. By studying the properties and characteristics of Sidon spaces, researchers can gain insights into the design and classification of optimum-distance cyclic subspace codes, leading to advancements in error correction coding theory.