Основні поняття
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.
Анотація
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.