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Let V be an n-dimensional vector space over the finite field F q and let V k q denote the family of all k-dimensional subspaces of V. A family F ⊆ V k q is called intersecting if for all F, F ′ ∈ F , we have dim (F ∩ F ′) ≥ 1. A family F ⊆ V k q is called almost intersecting if for every F ∈ F there is at most one element F ′ ∈ F satisfying dim (F ∩ F ′) = 0. In this paper we investigate almost intersecting families in the vector space V. Firstly, for large n, we determine the maximum size of an almost intersecting family in V k q , which is the same as that of an intersecting family. Secondly, we characterize the structures of all maximum almost intersecting families under the condition that they are not intersecting. [ABSTRACT FROM AUTHOR]

Liu et al. (IEEE Trans Inf Theory 68:3096–3107, 2022) investigated a class of BCH codes C (q , q + 1 , δ , 1) with q = δ m a prime power and proved that the set B δ + 1 of supports of the minimum weight codewords supports a Steiner system S (3 , δ + 1 , q + 1) . In this paper, we give an equivalent formulation of B δ + 1 in terms of elementary symmetric polynomials and then construct a number of mutually disjoint Steiner systems S (3 , δ + 1 , δ m + 1) when m is even and a number of mutually disjoint G-designs G ( δ m + 1 δ + 1 , δ + 1 , δ + 1 , 3) when m is odd. In particular, the existence of three mutually disjoint Steiner systems S (3 , 5 , 4 m + 1) or three mutually disjoint G-designs G ( 4 m + 1 5 , 5 , 5 , 3) is established. [ABSTRACT FROM AUTHOR]

A t-spontaneous emission error design, denoted by t-(v, k; m) SEED or t-SEED in short, is a system B of k-subsets of a v-set V with a partition B 1 , B 2 , … , B m of B satisfying | { B ∈ B i : E ⊆ B } | | B i | = μ E for any 1 ≤ i ≤ m and E ⊆ V , | E | ≤ t , where μ E is a constant depending only on E. A t-(v, k; m) SEED is an important combinatorial object with applications in quantum jump codes. The number m is called the dimension of the t-SEED and this corresponds to the number of orthogonal basis states in a quantum jump code. For given v, k and t, a t-(v, k; m) SEED is called optimal when m achieves the largest possible dimension. When k ∣ v , an optimal 1-(v, k; m) SEED has dimension v - 1 k - 1 and can be constructed by Baranyai’s Theorem. This note investigates optimal 1-(v, k; m) SEEDs with k ∤ v , in which a generalization of Baranyai’s Theorem plays a significant role. To be specific, we construct an optimal 1-(v, k; m) SEED for all positive integers v, k, s with v ≡ - s (mod k), k ≥ s + 1 and v ≥ max { 2 k , s (2 k - 1) } . [ABSTRACT FROM AUTHOR]

Objective: To investigate the occurrence and development of gastric mucosal atrophic lesions and their histopathological characteristics. Methods: Histopathological diagnosis and immunohistochemical staining using the EnVision two-step method were conducted on 1969 gastric mucosal atrophic lesions obtained from gastroscopic biopsy specimens. A total of 48-month three-stage endoscopic biopsy follow-ups were performed. Results: When the gastric mucosal epithelium was affected by infection, chemical irritation, or immune or genetic factors, the gastric mucosal epithelium glands atrophied, the mucosa became thinner, the number of glands decreased, the intestinal epithelium progressed to metaplasia and smooth muscle fibre became hyperplasia. Such changes may lead to the proliferation and dysplasia of epithelial cells of the gastric mucosa and neoplastic hyperplasia in nature; this is referred to as gastric mucosal atrophic lesions in this study. According to this definition, the present study divided gastric mucosal atrophy into four types: (1) glandular atrophy of the lamina propria; (2) compensatory proliferative atrophy; (3) intestinal metaplasia atrophy; and (4) smooth muscle proliferative atrophy. The incidence rates of the above were 40.1% (789/1969), 14.3% (281/1969), 27.8% (547/1969) and 17.9% (352/1969), respectively. One- to 4-year follow-ups found that the changes were not significant and that the percentages of patients with disease exacerbation were 85.7% (1688/1969) and 9.8% (192/1969). The percentages of patients who developed low-grade intraepithelial neoplasia and high-grade intraepithelial neoplasia were 2.8% (55/1969) and 1.1% (21/1969), respectively; 0.7% (13/1969) of patients developed intramucosal cancer. Conclusion: Gastric mucosal atrophic lesions and histopathological staging are based on the morphological characteristics of gastric mucosal atrophy and the hypothesis of malignant transformation of cells during the occurrence and development of mucosal atrophy. Mastering pathological staging is beneficial to clinicians for enacting precise treatment and is important for reducing the incidence of gastric cancer. [ABSTRACT FROM AUTHOR]

For distributed storage based on data replication, an optimal data placement is vital to minimize the variance of the number of available files. Well-balanced triple systems and nearly well-balanced triple systems (NWBTSs) were proposed to produce optimal data placements for triple replication. This article concentrates on the existence of NWBTSs. By constructing candelabra systems with various desirable partitions, many new constructions for NWBTSs are developed. The main result of this paper is that there always exist optimal data placements for triple replication on a v-set for all positive integers v possibly except when v ≡ 4 (mod 24) or v = 50 , 74 . [ABSTRACT FROM AUTHOR]

Studying the changes in the landscape pattern of coastal cities and analyzing their land use conditions are conducive to understanding the internal structure of the city and in-depth analysis of the law of urban development, so as to propose governance measures and improvement methods in the aspects of economy, people's livelihood, and environment. Based on geographic information system and machine learning technology, this paper analyzes the internal mechanism, temporal and spatial characteristics and change laws of the intensive use of sea areas. The selection of research scale is in the exploratory stage, which is based on the research scale of land intensive use. Moreover, this paper combines landscape ecology to construct a coastal city landscape image simulation system and uses remote sensing technology to analyze the system model. Finally, this paper analyzes the performance of the coastal city landscape image simulation system constructed in this paper through experimental analysis. From the research results, the system constructed in this paper basically meets the needs of coastal city landscape image simulation. [ABSTRACT FROM AUTHOR]

Large sets of combinatorial designs has always been a fascinating topic in design theory. These designs form a partition of the whole space into combinatorial designs with the same parameters. In particular, a large set of block designs, whose blocks are of size k taken from an n-set, is a partition of all the k-subsets of the n-set into disjoint copies of block designs, defined on the n-set, and with the same parameters. The current most intriguing question in this direction is whether large sets of Steiner quadruple systems exist and to provide explicit constructions for those parameters for which they exist. In view of its difficulty no one ever presented an explicit construction even for one nontrivial order. Hence, we seek for related generalizations. As generalizations, to the existence question of large sets, we consider two related questions. The first one is to provide constructions for sets on Steiner systems in which each block (quadruple or a k-subset) is contained in exactly μ systems. The constructions of such systems also yield secure protocols for the generalized Russian cards problem. The second question is to provide constructions for large set of H-designs (mainly for quadruples, but also for larger block size), which have applications in threshold schemes and in quantum jump codes. We prove the existence of such systems for many parameters using orthogonal arrays, perpendicular arrays, ordered designs, sets of permutations, and one-factorizations of the complete graph. [ABSTRACT FROM AUTHOR]

Various kinds of fingerprinting codes and their related combinatorial structures are extensively studied for protecting copyrighted materials. This paper concentrates on one specialised fingerprinting code named wide-sense frameproof codes in order to prevent innocent users from being framed. Let Q be a finite alphabet of size q. Given a t-subset X = { x 1 , ... , x t } ⊆ Q n , a position i is called undetectable for X if the values of the words of X match in their ith position: x i 1 = ⋯ = x i t . The wide-sense descendant set of X is defined by wdesc (X) = { y ∈ Q n : y i = x i 1 , i ∈ U (X) } , where U(X) is the set of undetectable positions for X. A code C ⊆ Q n is called a wide-sense t-frameproof code if wdesc (X) ∩ C = X for all X ⊆ C with | X | ≤ t . The paper improves the upper bounds on the sizes of wide-sense 2-frameproof codes by applying techniques on non 2-covering Sperner families and intersecting families in extremal set theory. [ABSTRACT FROM AUTHOR]

Quantum jump codes are quantum codes which correct errors caused by quantum jumps. A t-spontaneous emission error design (t-SEED) was introduced by Beth et al. in 2003 to construct quantum jump codes. The number of designs (dimension) in a t-SEED corresponds to the number of orthogonal basis states in a quantum jump code. Denote by M ¯ (t , k , v) the largest possible dimension m for which a nondegenerate t-(v, k; m) SEED exists. M ¯ (2 , 3 , v) has been determined completely, which is based on a great deal of research on large sets of Steiner triple systems and large sets of pairwise disjoint compatible 2-(v, 3, 1) packings. For k = 4 , the upper bounds on dimensions of 2-(v, 4; m) SEEDs were also demonstrated and the corresponding leave graphs were investigated in Zhou and Chang (J Combin Des 24:439–460, 2016). In this paper we turn our attention to the case t = 2 and k = 5 . We present general upper bounds on the dimensions of 2-(v, 5; m) SEEDs and describe the concrete leave graphs of the 2-SEEDs attaining the stated upper bounds. [ABSTRACT FROM AUTHOR]

An optical orthogonal code (OOC) is a family of binary sequences having good auto- and cross-correlation properties. Let Φ (v , k , λ a , λ c) denote the largest possible size among all (v , k , λ a , λ c) -OOCs. A (v , k , λ a , λ c) -OOC with Φ (v , k , λ a , λ c) codewords is said to be maximal. In this paper, we research into maximal (v , k , k - 2 , k - 1) -OOCs and determine the exact value of Φ (v , k , k - 2 , k - 1) . This generalizes the result on the special case of k = 4 by Huang and Chang in 2012. Distributions of differences with maximum multiplicity are analyzed by several classes to deal with the general case for all possible v and k. [ABSTRACT FROM AUTHOR]

An H-design H(v, g, k, t) is a triple (X , G , B) , where X is a set of gv points, G is a partition of X into v disjoint groups of size g, and B is a set of G -transverse k-subsets, called blocks, such that each G -transverse t-subset is contained in exactly one block of B . A frame-derived H-design FDH(v, g, 4, 3) is an H(v, g, 4, 3) whose derived design at every point forms a Kirkman frame, an H (v - 1 , g , 3 , 2) having a resolution into g (v - 1) / 2 holey parallel classes. FDH(v, g, 4, 3)s are proved to be effective designs to produce large sets of Kirkman triple systems (LKTS) in this paper. Related recursive constructions are established and direct constructions for FDH(v, 2, 4, 3)s are displayed by using automorphism groups. The existence result on LKTS is improved as well. [ABSTRACT FROM AUTHOR]

Research on the existence of large sets of Kirkman triple systems (LKTS) extends from the mid-eighteen hundreds to the present. Enlightened by known direct constructions of LKTSs, we bring forth new approaches and finally establish the existence of LKTSs of all admissible prime power sizes less than 400 only with two possible exceptions. In the process, we also employ known construction methods and draw support from efficient algorithms. [ABSTRACT FROM AUTHOR]

The combinatorial aspects of quantum codes were demonstrated in the study of decay processes of certain quantum systems used in the newly emerging field of quantum computing. Among them, the configuration of t-spontaneous emission error design ( t-SEED) was proposed to correct errors caused by quantum jumps. The number of designs (dimension) in a t-SEED corresponds to the number of orthogonal basis states in a quantum jump code. In this paper the upper and lower bounds on the dimensions of 3- $$(v,\,4;\,m)$$ SEEDs are studied and the necessary and sufficient conditions for 3-SEEDs attaining the upper bounds are described. Several new combinatorial constructions are presented for general t-SEEDs and lots of t-SEEDs of new parameters with $$t\ge 3$$ are shown to exist. [ABSTRACT FROM AUTHOR]

Research on the existence of large sets of Kirkman triple systems (LKTS) extends from the mid-eighteen hundreds to the present. In this paper we review known direct approaches of constructing LKTS and present new ideas of direct constructions. We finally prove the existence of an LKTS( v) where $$v \in \{69,141,165,213,285,309,333\}$$ . Combining recursive constructions yields several new infinite classes. [ABSTRACT FROM AUTHOR]

Quantum jump codes (JC) are quantum codes which correct errors caused by quantum jumps. A spontaneous emission error design (SEED) has been introduced to construct quantum JC. In this paper the nonexistence of some $$(n,m,t)_k$$ -JCs is proved, where $$m={n-t\atopwithdelims ()k-t},\, k\ge t+1,\, \text{ and }\, (n,t,k)=(2k+1,1,k), (2k,2,k),(7,2,3),(8,3,4)$$ . It is also shown that, for these parameters with $$t=1,2$$ , the existence of an $$(n,m,t)_k$$ -JC is equivalent to that of a $$t-(n,k;m)$$ -SEED. [ABSTRACT FROM AUTHOR]

An LPDTS( ν) is a collection of 3( ν − 2) disjoint pure directed triple systems on the same set of ν elements. It is showed in Tian’s doctoral thesis that there exists an LPDTS( gn) for gn ≡ 0,4 (mod 6), ν ⩾ 4. In this paper, we establish the existence of an LPDTS( ν) for ν ≡ 1,3 (mod 6), ν > 3. Thus the spectrum for LPDTS( ν) is completely determined to be the set { ν:ν ≡ 0, 1 (mod 3), ν ⩾ 4}. [ABSTRACT FROM AUTHOR]