Embedding in a perfect code

A binary 1-error-correcting code can always be embedded in a 1-perfect code of some larger lengthComment: Eng: 5pp, Rus: 5pp. V3: revised, a survey added; the accepted version; Russian translation adde

Completion and deficiency problems

Given a partial Steiner triple system (STS) of order $n$, what is the order of the smallest complete STS it can be embedded into? The study of this question goes back more than 40 years. In this paper we answer it for relatively sparse STSs, showing that given a partial STS of order $n$ with at most $r \le \varepsilon n^2$ triples, it can always be embedded into a complete STS of order $n+O(\sqrt{r})$, which is asymptotically optimal. We also obtain similar results for completions of Latin squares and other designs. This suggests a new, natural class of questions, called deficiency problems. Given a global spanning property $\mathcal{P}$ and a graph $G$, we define the deficiency of the graph $G$ with respect to the property $\mathcal{P}$ to be the smallest positive integer $t$ such that the join $G\ast K_t$ has property $\mathcal{P}$. To illustrate this concept we consider deficiency versions of some well-studied properties, such as having a $K_k$-decomposition, Hamiltonicity, having a triangle-factor and having a perfect matching in hypergraphs. The main goal of this paper is to propose a systematic study of these problems; thus several future research directions are also given

The Intersection problem for 2-(v; 5; 1) directed block designs

The intersection problem for a pair of 2-(v, 3, 1) directed designs and 2-(v, 4, 1) directed designs is solved by Fu in 1983 and by Mahmoodian and Soltankhah in 1996, respectively. In this paper we determine the intersection problem for 2-(v, 5, 1) directed designs.Comment: 17 pages. To appear in Discrete Mat

On the geometry of full points of abstract unitals

The concept of full points of abstract unitals has been introduced by Korchm\'aros, Siciliano and Sz\H{o}nyi as a tool for the study of projective embeddings of abstract unitals. In this paper we give a more detailed description of the combinatorial and geometric structure of the sets of full points in abstract unitals of finite order

Exploring Link Prediction over Hyper-Relational Temporal Knowledge Graphs Enhanced with Time-Invariant Relational Knowledge

Stemming from traditional knowledge graphs (KGs), hyper-relational KGs (HKGs) provide additional key-value pairs (i.e., qualifiers) for each KG fact that help to better restrict the fact validity. In recent years, there has been an increasing interest in studying graph reasoning over HKGs. In the meantime, due to the ever-evolving nature of world knowledge, extensive parallel works have been focusing on reasoning over temporal KGs (TKGs), where each TKG fact can be viewed as a KG fact coupled with a timestamp (or time period) specifying its time validity. The existing HKG reasoning approaches do not consider temporal information because it is not explicitly specified in previous benchmark datasets. Besides, all the previous TKG reasoning methods only lay emphasis on temporal reasoning and have no way to learn from qualifiers. To this end, we aim to fill the gap between TKG reasoning and HKG reasoning. We develop two new benchmark hyper-relational TKG (HTKG) datasets, i.e., Wiki-hy and YAGO-hy, and propose a HTKG reasoning model that efficiently models both temporal facts and qualifiers. We further exploit additional time-invariant relational knowledge from the Wikidata knowledge base and study its effectiveness in HTKG reasoning. Time-invariant relational knowledge serves as the knowledge that remains unchanged in time (e.g., Sasha Obama is the child of Barack Obama), and it has never been fully explored in previous TKG reasoning benchmarks and approaches. Experimental results show that our model substantially outperforms previous related methods on HTKG link prediction and can be enhanced by jointly leveraging both temporal and time-invariant relational knowledge

Self-embeddings of Hamming Steiner triple systems of small order and APN permutations

The classification, up to isomorphism, of all self-embedding monomial power permutations of Hamming Steiner triple systems of order n = 2 m − 1 for small m (m ≤ 22), is given. As far as we know, for m ∈ {5, 7, 11, 13, 17, 19}, all given self-embeddings in closed surfaces are new. Moreover, they are cyclic for all m and nonorientable at least for all m ≤ 19. For any non prime m, the nonexistence of such self-embeddings in a closed surface is proven. The rotation line spectrum for self-embeddings of Hamming Steiner triple systems in pseudosurfaces with pinch points as an invariant to distinguish APN permutations or, in general, to classify permutations, is also proposed. This invariant applied to APN monomial power permutations gives a classification which coincides with the classification of such permutations via CCZ-equivalence, at least up to m ≤ 17