On two non-existence results for Cameron–Liebler k-sets in \({{\,\mathrm{\textrm{PG}}\,}}(n,q)\)

The article delves into the intricate world of Cameron–Liebler k-sets, focusing on non-existence results and modular equalities in both projective and affine spaces. It builds upon previous research by disproving conjectures and discovering new non-trivial examples, while also providing lower bounds and modular conditions for the parameter x. The paper highlights the rare occurrence of non-trivial examples and the significance of non-existence results, particularly in large dimensions. The authors employ various techniques, including induction and projections, to improve existing bounds and extend known results to higher dimensions. The work is essential for researchers aiming to classify and understand the properties of Cameron–Liebler sets, contributing to the broader field of algebraic coding theory and combinatorial designs.