Existence of quasi-optimal cyclic orbit codes in the remaining parameter cases

Determine whether quasi-optimal cyclic orbit codes exist for the parameter regimes not covered by Theorem 6.7 and Theorem 4.5, namely cases other than even-dimensional ambient spaces or generator dimension \(k=3\).

Background

Theorem 6.7 establishes the existence of quasi-optimal full-length cyclic orbit codes when the ambient dimension $n$ is even, while Theorem 4.5 provides a classification and existence result for generator dimension $k=3$.

The paper does not settle existence in the remaining parameter regimes. The authors identify this unresolved existence question explicitly as an open problem.