Constructions of Strictlym-Cyclic and Semi-Cyclic H(m,n,4,3)

Abstract

An H(m, n, 4, 3) is a triple (X, T , B), where X is a set of mn points, T is a partition of X into m disjoint sets of size n and B is a set of 4-element transverses of T , such that each 3-element transverse of T is contained in exactly one of them. If the full automorphism group of an H(m, n, 4, 3) admits an automorphism α consisting of n cycles of length m (resp. m cycles of length n), then this H(m, n, 4, 3) is called m-cyclic (resp. semi-cyclic). Further, if all blockorbits of an m-cyclic (resp. semi-cyclic) H(m, n, 4, 3) are full, then it is called strictly cyclic. In this paper, we construct some infinite classes of strictly m-cyclic and semi-cyclic H(m, n, 4, 3), and use them to give new infinite classes of perfect two-dimensional optical orthogonal codes with maximum collision parameter λ = 2 and AM-OPPTS/AM-OPPW property.

Holographic Entropy Relations

“…Although, as we showed, it is straightforward to construct quantum states that realize the vectors which generate all the extremal rays of the bipartite entropy cone, it is not true that an arbitrary conical combination (viz., a linear combination with non-negative coefficients) of these vectors can be exactly realized by a quantum state. In particular it is important to notice that the holographic entropy cone [22] 9 By definition, these are one-dimensional subspaces of the entropy space -they are simply rays emanating from the origin which generate the polyhedral cone. In particular, any vector within the cone can be obtained as a conical combination of the extremal rays.…”

“…However, since in quantum field theory the von Neumann entropy is generically infinite, the interpretation of this vector is unclear. One possibility 10 While the authors of [22] were interested in holographic field theories where entanglement is plagued by UV divergences, finiteness was achieved by considering states in the tensor product of a set of holographic field theories. Geometrically these states correspond to multi-boundary wormhole geometries, and by restricting the allowed subsystems to be entire boundaries, one has finite entanglement (per unit spatial volume).…”

“…The situation is more dire for the other inequalities explored in the context of the holographic entropy cone. [22] For instance, these authors proved a set of inequalities for five subsystems (and a family of inequalities for a higher number of parties). However, in their present form these inequalities do not render themselves to a simple physical interpretation.…”

“…[19] Two further (distinct) proofs of MMI based on the 'bit thread' reformulation of holographic entanglement entropy [20] recently appeared in [17,21]. these inequalities hold for general time-dependent geometries, since the analysis of [22] was restricted to time-reflection symmetric states where the RT prescription can be applied. 7 So far, we could at best realize that certain specific combinations of entanglement entropies have a definite sign, but hitherto we did not have a good way of deriving further inequalities, or, for the ones which are known, the corresponding information quantities directly.…”

“…In the holographic context our framework is complementary to the holographic entropy cone construction of [22], as we further explain below: rather than focusing on the entropy vectors, we work with entropy relations (corresponding to hyperplanes 7 One can argue that this restriction can be lifted in the case of twodimensional conformal field theories with AdS 3 holographic duals. [23] We thank Xi Dong for discussions on this subject.…”