Related papers: On $2$-parent-identifying set systems of block siz…
Two novel views are presented on the trapdoor channel. First, by deriving the underlying iterated function system (IFS), it is shown that the trapdoor channel with input blocks of length $n$ can be regarded as the $n$th element of a…
We construct asymptotically optimal adjacency labelling schemes for every hereditary class containing $2^{\Omega(n^2)}$ $n$-vertex graphs as $n\to \infty$. This regime contains many classes of interest, for instance perfect graphs or…
Subspace codes are collections of subspaces of a projective space such that any two subspaces satisfy a pairwise minimum distance criterion. Recent results have shown that it is possible to construct optimal $(5,3)$ subspace codes from…
Identification of latent binary sequences from a pool of noisy observations has a wide range of applications in both statistical learning and population genetics. Each observed sequence is the result of passing one of the latent…
It is shown that the maximum size of a binary subspace code of packet length $v=6$, minimum subspace distance $d=4$, and constant dimension $k=3$ is $M=77$; in Finite Geometry terms, the maximum number of planes in $\operatorname{PG}(5,2)$…
Splitter sets have been widely studied due to their applications in flash memories, and their close relations with lattice tilings and conflict avoiding codes. In this paper, we give necessary and sufficient conditions for the existence of…
In combinatorial game theory, the lower and upper bounds of the number of games born by day $4$ have been recognized as $3.0 \cdot 10^{12}$ and $10^{434}$, respectively. In this study, we improve the lower bound to $10^{28.2}$ and the upper…
Codes over trees were introduced recently to bridge graph theory and coding theory with diverse applications in computer science and beyond. A central challenge lies in determining the maximum number of labelled trees over $n$ nodes with…
Since ancient times mathematicians consider geometrical objects with integral side lengths. We consider plane integral point sets $\mathcal{P}$, which are sets of $n$ points in the plane with pairwise integral distances where not all the…
In connection with optimal state determination for two qubits, the question was raised about the maximum number of pairwise complementary reductions. The main result of the paper tells that the maximum number is 4, that is, if A1, A2,... Ak…
Domination of grids has been proved to be a demanding task and with the addition of independence it becomes more challenging. It is known that no grid with $m,n \geq 5$ has an efficient dominating set, also called perfect code, that is, an…
A finite set $P$ of points in the plane is $n$-universal with respect to a class $\mathcal{C}$ of planar graphs if every $n$-vertex graph in $\mathcal{C}$ admits a crossing-free straight-line drawing with vertices at points of $P$. For the…
In a graph $G$, a set $D\subseteq V(G)$ is called 2-dominating set if each vertex not in $D$ has at least two neighbors in $D$. The 2-domination number $\gamma_2(G)$ is the minimum cardinality of such a set $D$. We give a method for the…
An $r$-identifying code in a graph $G = (V,E)$ is a subset $C \subseteq V$ such that for each $u \in V$ the intersection of $C$ and the ball of radius $r$ centered at $u$ is nonempty and unique. Previously, $r$-identifying codes have been…
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,…
In this note, we study the classification of $\mathbb{Z}_4$-codes. For some special cases $(k_1,k_2)$, by hand, we give a classification of $\mathbb{Z}_4$-codes of length $n$ and type $4^{k_1}2^{k_2}$ satisfying a certain condition. Our…
Let $A(n, d)$ denote the maximum size of a binary code of length $n$ and minimum Hamming distance $d$. Studying $A(n, d)$, including efforts to determine it as well to derive bounds on $A(n, d)$ for large $n$'s, is one of the most…
Optical orthogonal signature pattern codes (OOSPCs) play an important role in a novel type of optical code-division multiple-access (CDMA) network for 2-dimensional image transmission. There is a one-to-one correspondence between an $(m, n,…
The Identifying Code (IC) problem seeks a vertex subset whose intersection with every vertex's closed neighborhood is unique, enabling fault detection in multiprocessor systems and practical uses in identity verification, environmental…
Batch codes, introduced by Ishai, Kushilevitz, Ostrovsky and Sahai, represent the distributed storage of an $n$-element data set on $m$ servers in such a way that any batch of $k$ data items can be retrieved by reading at most one (or more…