Let φ : C2×C2 → C, φ((x1, x2), (y1, y2)) = (x1−y1)2+(x2−y2)2. We say that f : C2 → C2 preserves distance d ≥ 0, if for each X, Y ∈ C2 φ(X, Y ) = d2 implies φ(f(X), f(Y )) = d2. We prove that each unit-distance preserving mapping f : C2 → C2 has a form I◦(γ, γ), where γ : C→ C is a field homomorphism and I : C2 → C2 is an affine mapping with orthogonal linear part. We prove an analogous result f...

For each bipartition_of a bipartite distance-regular graph F, there naturally corresponds another distance-regular graph F called a halved graph. It is shown that the existence of a perfect e-code in a halved graph F is equivalent to the existence of a uniformly packed 2e-code in F with certain specific parameters. Using this equivalence, we show the non-existence of perfect codes for two class...

Topochemical reactions are a promising method to obtain crystalline polymeric materials with distance-determined regio- or stereoselectivity. It has been concluded on an empirical basis that the closest intermolecular C⋯C distance in crystals of alkynes, d(C⋯C)min, should reach threshold ∼3 Å for bonding occur at room temperature. To understand this threshold, we study here polymerization acety...

The square C of a linear error correcting code C is the linear code spanned by the component-wise products of every pair of (non-necessarily distinct) words in C. Squares of codes have gained attention for several applications mainly in the area of cryptography, and typically in those applications one is concerned about some of the parameters (dimension, minimum distance) of both C and C. In th...

For a code C, bounded-distance decoding algorithms perform as optimal algorithms within the balls B(c), centered at the codewords c 2 C, with radius equal to half the minimum Euclidean distance of the code. Thus distinct bounded-distance algorithms vary in performance due to their different behavior outside the balls B(c). We investigate this issue by analyzing the decision regions of some know...

The quasispecies model introduced by Eigen in 1971 has close connections with the isometry group of the space of binary sequences relative to the Hamming distance metric. Generalizing this observation we introduce an abstract quasispecies model on a finite metric space X together with a group of isometries Γ acting transitively on X . We show that if the domain of the fitness function has a nat...

the emph{harary index} $h(g)$ of a connected graph $g$ is defined as $h(g)=sum_{u,vin v(g)}frac{1}{d_g(u,v)}$ where $d_g(u,v)$ is the distance between vertices $u$ and $v$ of $g$. the steiner distance in a graph, introduced by chartrand et al. in 1989, is a natural generalization of the concept of classical graph distance. for a connected graph $g$ of order at least $2$ and $ssubseteq v(g)$, th...

a set $s$ of vertices in a graph $g=(v,e)$ is called a total$k$-distance dominating set if every vertex in $v$ is withindistance $k$ of a vertex in $s$. a graph $g$ is total $k$-distancedomination-critical if $gamma_{t}^{k} (g - x) < gamma_{t}^{k}(g)$ for any vertex $xin v(g)$. in this paper,we investigate some results on total $k$-distance domination-critical of graphs.