From cosets of binary Hamming codes we construct diameter perfect constant-weight ternary codes with weight $n-1$ (where $n$ is the code length) and distances 3 and 5. The class of distance 5 codes has parameters unknown before. Keywords: constant-weight codes, ternary codes, perfect codes, diameter perfect codes, perfect matchings, Preparata codes
In 1968, Golomb and Welch conjectured that there is no perfect Lee codes with radius $r\ge2$ and dimension $n\ge3$. A diameter perfect code is a natural generalization of the perfect code. In 2011, Etzion (IEEE Trans. Inform. Theory, 57(11): 7473--7481, 2011) proposed the following problem: Are there diameter perfect Lee (DPL, for short) codes with diameter greater than four besides the $DPL(3,6)$ code? Later, Horak and AlBdaiwi (IEEE Trans. Inform. Theory, 58(8): 5490--5499, 2012) conjectured that there are no $DPL(n,d)$ codes for dimension $n\ge3$ and diameter $d>4$ except for $(n,d)=(3,6)$.…
Diameter perfect codes form a natural generalization for perfect codes. They are based on the code-anticode bound which generalizes the sphere-packing bound. The code-anticode bound was proved by Delsarte for distance-regular graphs and it holds for some other metrics too. In this paper we prove the bound for non-binary constant-weight codes with the Hamming metric and characterize the diameter perfect codes and the maximum size anticodes for these codes. We distinguish between six families of non-binary diameter constant-weight codes and four families of maximum size non-binary constant-weigh…
Perfect Matching-Cut is the problem of deciding whether a graph has a perfect matching that contains an edge-cut. We show that this problem is NP-complete for planar graphs with maximum degree four, for planar graphs with girth five, for bipartite five-regular graphs, for graphs of diameter three and for bipartite graphs of diameter four. We show that there exist polynomial time algorithms for the following classes of graphs: claw-free, $P_5$-free, diameter two, bipartite with diameter three and graphs with bounded tree-width.
Perfect codes in the $n$-dimensio\-nal grid $Λ_n$ of the lattice $\mathbb{Z}^n$ ($0<n\in\mathbb{Z}$) and its quotient toroidal grids were obtained via the truncated distance in $\mathbb{Z}^n$ given between $u=(u_1,\cdots,u_n)$ and $v=(v_1, \ldots,v_n)$ as the graph distance $h(u,v)$ in $Λ_n$, if $|u_i-v_i|\le 1$, for all $i\in\{1, \ldots,n\}$, and as $n+1$, otherwise. Such codes are extended to multilattice graphs $Γ_n$ obtained by glueing ternary $n$-cubes along their codimension 1 ternary subcubes in such a way that each binary $n$-subcube is contained in a unique maximal lattice of $Γ_n$. T…
We study a family of graphs with diameter two and asymptotically optimal order for their maximum degree, obtained from perfect difference sets. We show that for all known examples of perfect difference sets, the graph we obtain is isomorphic to one of the Brown graphs, a well-known family of graphs in the degree-diameter problem.
In 1968, Golomb and Welch conjectured that there does not exist perfect Lee code in $\mathbb{Z}^{n}$ with radius $r\ge2$ and dimension $n\ge3$. Besides its own interest in coding theory and discrete geometry, this conjecture is also strongly related to the degree-diameter problems of abelian Cayley graphs. Although there are many papers on this topic, the Golomb-Welch conjecture is far from being solved. In this paper, we prove the nonexistence of linear perfect Lee codes by introducing some new algebraic methods. Using these new methods, we show the nonexistence of linear perfect Lee codes of…
Geometrical objects with integral sides have attracted mathematicians for ages. For example, the problem to prove or to disprove the existence of a perfect box, that is, a rectangular parallelepiped with all edges, face diagonals and space diagonals of integer lengths, remains open. More generally an integral point set $\mathcal{P}$ is a set of $n$ points in the $m$-dimensional Euclidean space $\mathbb{E}^m$ with pairwise integral distances where the largest occurring distance is called its diameter. From the combinatorial point of view there is a natural interest in the determination of the s…
Let $G$ be a connected graph with $V(G)=\{v_1,\ldots,v_n\}$. The $(i,j)$-entry of the distance matrix $D(G)$ of $G$ is the distance between $v_i$ and $v_j$. In this article, using the well-known Ramsey's theorem, we prove that for each integer $k\ge 2$, there is a finite amount of graphs whose distance matrices have rank $k$. We exhibit the list of graphs with distance matrices of rank $2$ and $3$. Besides, we study the rank of the distance matrices of graphs belonging to a family of graphs with their diameters at most two, the trivially perfect graphs. We show that for each $η\ge 1$ there exi…
