On the distance distributions of single-orbit cyclic subspace codes

TL;DR

This paper investigates the intersection distributions of single-orbit cyclic subspace codes, revealing divisibility properties of codeword pair counts based on the orbit stabilizer and subspace structure.

Contribution

It proves new divisibility results for intersection counts in cyclic subspace codes with specific stabilizers and subspace properties, extending understanding of their distance distributions.

Findings

Number of codeword pairs with a given intersection dimension is divisible by q^t(q^t+1) under certain conditions.

For even n/t, the count of pairs with intersection dimension 2tm depends on the number of cyclic shifts contained in the subspace.

Examples illustrate the theoretical divisibility properties and intersection distributions.

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 $(U,\alpha U)$ such that $\dim(U\cap \alpha U)=i$ 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 $(U,\alpha U)$ 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 $(U,\alpha U)$ 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.