Distance-distribution rigidity of the orbit codes \(\mathcal{O}_{s,y}\)

Prove or disprove the conjecture that, for small parameters \(q\) and \(n=2k\), the distance distribution of the cyclic orbit codes \(\mathcal{O}_{s,y}=\operatorname{Orb}(U_{s,y})\) depends only on \(n\), \(q\), and whether the generating subspace \(U_{s,y}\) contains a cyclic shift of the subfield \(\mathbb{F}_{q^2}\), equivalently that the parameter \(r\) in Theorem 5.7 is determined solely by these data.

Background

The paper studies the family of cyclic orbit codes $\mathcal{O}_{s,y}=\operatorname{Orb}(U_{s,y})$ in the case $n=2k$, where $U_{s,y}=\{u+u^{q^s}y:u\in\mathbb{F}_{q^k}\}$. For quasi-optimal codes, Theorem 5.7 describes possible intersection and distance distributions using an additional integer parameter $r$.

Computations for small values of $q$ and $n=2k$ suggest that this parameter is rigid: it may be determined by the ambient parameters and by whether $U_{s,y}$ contains a cyclic shift of $\mathbb{F}_{q^2}$. The authors explicitly leave open whether this observed pattern holds generally or can be disproved.