Related papers: Additive perfect codes in Doob graphs

A subset $C$ of the vertex set of a graph $\Gamma$ is said to be $(\alpha,\beta)$-regular if $C$ induces an $\alpha$-regular subgraph and every vertex outside $C$ is adjacent to exactly $\beta$ vertices in $C$. In particular, if $C$ is an…

For a nonzero integer n, a set of m distinct nonzero integers {a_1,a_2,...,a_m} such that a_i a_j + n is a perfect square for all 1 <= i < j <= m, is called a D(n)-m-tuple. In this paper, by using properties of so-called regular Diophantine…

We show that (n,2^n) additive codes over GF(4) can be represented as directed graphs. This generalizes earlier results on self-dual additive codes over GF(4), which correspond to undirected graphs. Graph representation reduces the…

A $k$-regular graph is called a divisible design graph (DDG for short) if its vertex set can be partitioned into $m$ classes of size $n$, such that two distinct vertices from the same class have exactly $\lambda_1$ common neighbors, and two…

Golomb and Welch conjectured in 1970 that there only exist perfect Lee codes for radius $t=1$ or dimension $n=1, 2$. It is admitted that the existence and the construction of quasi-perfect Lee codes have to be studied since they are the…

Quantum stabilizer states over GF(m) can be represented as self-dual additive codes over GF(m^2). These codes can be represented as weighted graphs, and orbits of graphs under the generalized local complementation operation correspond to…

Let $G$ be a finite group and $m$ be an integer. We employ the notation $g_i$ to represent elements $(g,i)$ in the Cartesian product $G \times \mathbb{Z}_m$, where $\mathbb{Z}_m$ denotes integers modulo $m$. For given sets $T_{i,j}…

An exact $(k,d)$-coloring of a graph $G$ is a coloring of its vertices with $k$ colors such that each vertex $v$ is adjacent to exactly $d$ vertices having the same color as $v$. The exact $d$-defective chromatic number, denoted…

In this paper, we construct additively graceful signed graphs S from a given graph G that may be additively graceful or not be additively graceful. We also show the construction of additively graceful signed graphs from additively graceful…

Let $\Z_n[i]$ be the ring of Gaussian integers modulo a positive integer $n$. Very recently, Camarero and Mart\'{i}nez [IEEE Trans. Inform. Theory, {\bf 62} (2016), 1183--1192], showed that for every prime number $p>5$ such that $p\equiv…

Let $G$ be a simple graph of order $n$. The domination polynomial of $G$ is the polynomial $D(G, x)=\sum_{i=1}^n d(G,i) x^i$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$. The $n$-barbell graph $Bar_n$ with $2n$…

A $k$-regular graph is called a divisible design graph (DDG for short) if its vertex set can be partitioned into $m$ classes of size $n$, such that two distinct vertices from the same class have exactly $\lambda_1$ common neighbors, and two…

In order to construct quantum $[[n,0,d]]$ codes for $(n,d)=(56,15)$, $(57,15)$, $(58,16)$, $(63,16)$, $(67,17)$, $(70,18)$, $(71,18)$, $(79,19)$, $(83,20)$, $(87,20)$, $(89,21)$, $(95,20)$, we construct self-dual additive…

Let $\mathbb{D}$ be a division ring, and let ${\mathbb{D}}^{m\times n}$ be the set of $m\times n$ matrices over $\mathbb{D}$. Two matrices $A,B\in {\mathbb{D}}^{m\times n}$ are adjacent if ${\rm rank}(A-B)=1$. By the adjacency,…

The study of optical orthogonal codes has been motivated by an application in an optical code-division multiple access system. This paper focuses on optimal two-dimensional optical orthogonal codes with autocorrelation and cross-correlation…

We consider extended $1$-perfect codes in Hamming graphs $H(n,q)$. Such nontrivial codes are known only when $n=2^k$, $k\geq 1$, $q=2$, or $n=q+2$, $q=2^m$, $m\geq 1$. Recently, Bespalov proved nonexistence of extended $1$-perfect codes for…

We study properties of binary codes with parameters close to the parameters of 1-perfect codes. An arbitrary binary $(n=2^m-3, 2^{n-m-1}, 4)$ code $C$, i.e., a code with parameters of a triply-shortened extended Hamming code, is a cell of…

For the modular lattice D^4 = {1+1+1+1} associated with the extended Dynkin diagram \tilde{D}_4 (and also for D^r, where r > 4), Gelfand and Ponomarev introduced the notion of admissible and perfect lattice elements and classified them. In…

For a pair of given binary perfect codes C and D of lengths t and m respectively, the Mollard construction outputs a perfect code M(C,D) of length tm + t + m, having subcodes C1 and D2, that are obtained from codewords of C and D…

For a finite group $G$, the proper power graph $\mathscr{P}^*(G)$ of $G$ is the graph whose vertices are non-trivial elements of $G$ and two vertices $u$ and $v$ are adjacent if and only if $u \neq v$ and $u^m=v$ or $v^m=u$ for some…