On the distance distributions of single-orbit cyclic subspace codes

{A cyclic subspace code is a union of the orbits of subspaces contained in it. In a recent paper, Gluesing-Luerssen et al. (Des. Codes Cryptogr. 89, 447-470, 2021) showed that the study of the distance distribution of a single orbit cyclic subspace code is equivalent to the study of its intersection distribution. In this paper we have proved that in the orbit of a subspace U of Fqⁿ that has the stabilizer Fqᵗ^*(t ≠ n), the number of codeword pairs (U,α U) such that (U∩ α U)=i for any i,~ 0≤ i < (U), is a multiple of qᵗ(qᵗ+1), if n/t is an odd number. In the case of even n/t, if U contains q²ᵗᵐ-1q²ᵗ-1~ (m≥ 0) distinct cyclic shifts of F_q²ᵗ, then the number of codeword pairs (U,α U) with intersection dimension $2tm$ is equal to qᵗ+rqᵗ(qᵗ+1), for some non-negative integer r; and the number of codeword pairs (U,α U) with intersection dimension i,~(i≠ 2tm) is a multiple of qᵗ(qᵗ+1). Some examples have been given to illustrate the results presented in the paper.