On the distance distributions of single-orbit cyclic subspace codes
Mahak, Maheshanand Bhaintwal
TL;DR
The paper addresses the distance distribution problem for single-orbit cyclic subspace codes by translating it into the intersection distribution of a base subspace $U$ under cyclic shifts. It proves divisibility properties for the counts of shifts with a given intersection dimension, showing that when the stabilizer is $\\mathbb{F}_{q^t}^*$ and $n/t$ is odd these counts are multiples of $q^t(q^t+1)$, with additional structure in the even $n/t$ case tied to cyclic shifts of $\\mathbb{F}_{q^{2t}}$. The results are extended to non-full-length orbits by leveraging the $\\ ext{over }\\mathbb{F}_{q^t}$-vector-space structure, producing analogous divisibility patterns and special cases. Computational examples illustrate the divisibility phenomena, and the work lays groundwork for analyzing more complex, multi-orbit codes and for refining Sidon-space approaches in orbit-code design.
Abstract
{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 $\mathbb{F}_{q^n}$ that has the stabilizer $\mathbb{F}_{q^t}^*(t \neq n)$, the number of codeword pairs $(U,αU)$ such that $\dim(U\cap αU)=i$ for any $i,~ 0\leq i < \dim(U)$, is a multiple of $q^t(q^t+1)$, if $\frac{n}{t}$ is an odd number. In the case of even $\frac{n}{t}$, if $U$ contains $\frac{q^{2tm}-1}{q^{2t}-1}~ (m\geq 0)$ distinct cyclic shifts of $\mathbb{F}_{q^{2t}}$, then the number of codeword pairs $(U,αU)$ with intersection dimension $2tm$ is equal to $q^t+rq^t(q^t+1)$, for some non-negative integer $r$; and the number of codeword pairs $(U,αU)$ with intersection dimension $i,~(i\neq 2tm)$ is a multiple of $q^t(q^t+1)$. Some examples have been given to illustrate the results presented in the paper.