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The reconfigurable intelligent surface (RIS) technology is a potential solution to enhance network capacity and coverage without significant investment in additional infrastructure in 6G networks. This work highlights the interest of the mobile communication industry in RIS, and discusses the development of liquid crystal-based RIS for improved energy efficiency and coverage in the millimeter-wave band. Furthermore, the paper discusses perspectives and insights from an industry R&D point of view, addressing relevant use cases, technical requirements, implementation challenges, and practical considerations for RIS deployment optimization in the context of 6G networks. A hardware design of an RIS with liquid crystal at 28 GHz is presented. A propagation model for RIS as a new part of the system architecture is discussed, with approaches of semi-empirical models, geometric models, and their combination through the application of artificial intelligence/machine learning. Finally, a channel model for deployment optimization and dimensioning is presented, with the findings that a rather large RIS is favorable for coverage improvement, as well as greater attenuation at higher frequencies combined with a smaller RIS size., Comment: Published in IEEE Access

We present a representation for permutation groups as the automorphism group of an ordered set $U$ such that the automorphism group's action on a subset $T\subseteq U$ is the permutation group itself. For many imprimitive permutation groups, the representation uses fewer points than the smallest representations to date.

This paper exploits adjacencies between the orbits of an ordered set P and a consequence of the classification of finite simple groups to, in many cases, exponentially bound the number of automorphisms. Results clearly identify the structures which currently prevent the proof of such an exponential bound, or which indeed inflate the number of automorphisms beyond such a bound. This is a first step towards a possible resolution of the Automorphism Conjecture for ordered sets.

We introduce a recursive method to deconstruct the automorphism group of an ordered set. By connecting this method with deep results for permutation groups, we prove the Automorphism Conjecture for ordered sets of width less than or equal to $11$. Subsequent investigations show that the method presented here could lead to a resolution of the Automorphism Conjecture., Comment: arXiv admin note: substantial text overlap with arXiv:2209.09312

We prove the automorphism conjecture for ordered sets of width less than or equal to 11. The proof supports the meta conjecture that a large number of automorphisms is achievable only as some type of product of independent automorphisms on highly symmetric subsets., Comment: 57 pages, 6 figures

We prove that there are arbitrarily large indecomposable ordered sets T with a 2-chain C such that the smallest indecomposable proper superset U of C in T is T itself. Subsequently, we characterize all such indecomposable ordered sets T and 2-chains C. We also prove the same type of result for 2-antichains., Comment: 12 pages, 3 figures

We prove that for any $n$ there is a pair $(P_1 ^n , P_2 ^n )$ of nonisomorphic ordered sets such that $P_1 ^n $ and $P_2 ^n $ have equal maximal and minimal decks, equal neighborhood decks, and there are $n+1$ ranks $k_0 , \ldots , k_n $ such that for each $i$ the decks obtained by removing the points of rank $k_i $ are equal. The ranks $k_1 , \ldots , k_n $ do not contain extremal elements and at each of the other ranks there are elements whose removal will produce isomorphic cards. Moreover, we show that such sets can be constructed such that only for ranks $1$ and $2$, both without extremal elements, the decks obtained by removing the points of rank $r_i $ are not equal., Comment: 30 pages, 6 figures, straight LaTeX