Related papers: On the existence of some completely regular codes …
In Cayley graphs on the additive group of a small vector space over GF$(q)$, $q=2,3$, we look for completely regular (CR) codes whose parameters are new in Hamming graphs over the same field. The existence of a CR code in such Cayley graph…
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…
In this paper we obtain the necessary condition for the existence of perfect $k$-colorings (equitable $k$-partitions) in Hamming graphs $H(n,q)$, where $q=2,3,4$ and Doob graphs $D(m,n)$. As an application, we prove the non-existence of…
We consider the problem of existence of perfect $2$-colorings in the Doob graphs $D(m,n)$ and $4$-ary Hamming graphs $H(n,4)$. We characterize all parameters for which multifold $1$-perfect code in $D(m,n)$ exists. Also, we prove that for…
We investigate the class of completely regular codes in graphs with a distance partition C_0,..., C_\rho, where each set C_i, for 0<=i<=r-1, is an independent set. This work focuses on the existence problem for such codes in the…
In this paper, we explore completely regular codes in the Hamming graphs and related graphs. Experimental evidence suggests that many completely regular codes have the property that the eigenvalues of the code are in arithmetic progression.…
We obtain a classification of the completely regular codes with covering radius 1 and the second eigenvalue in the Hamming graphs H(3,q) up to q and intersection array. Due to works of Meyerowitz, Mogilnykh and Valyuzenich, our result…
Given a parity-check matrix $H_m$ of a $q$-ary Hamming code, we consider a partition of the columns into two subsets. Then, we consider the two codes that have these submatrices as parity-check matrices. We say that anyone of these two…
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…
We consider the problem of existence of perfect $2$-colorings (equitable $2$-partitions) of Hamming graphs with given parameters. We start with conditions on parameters of graphs and colorings that are necessary for their existence. Next we…
In this paper we consider completely regular codes, obtained from perfect (Hamming) codes by lifting the ground field. More exactly, for a given perfect code C of length n=(q^m-1)/(q-1) over F_q with a parity check matrix H_m, we define a…
In this paper infinite families of linear binary nested completely regular codes are constructed. They have covering radius $\rho$ equal to $3$ or $4$, and are $1/2^i$-th parts, for $i\in\{1,\ldots,u\}$ of binary (respectively, extended…
Let L be a Desarguesian 2-spread in the Grassmann graph $J_q(n,2)$. We prove that the collection of the 4-subspaces, which do not contain subspaces from L is a completely regular code in $J_q(n,4)$. Similarly, we construct a completely…
A perfect code in a graph $\Gamma = (V, E)$ is a subset $C$ of $V$ that is an independent set such that every vertex in $V \setminus C$ is adjacent to exactly one vertex in $C$. A total perfect code in $\Gamma$ is a subset $C$ of $V$ such…
We consider constructions of covering-radius-1 completely regular codes, or, equivalently, equitable 2-partitions (regular 2-partitions, perfect 2-colorings), of halved n-cubes. Keywords: completely regular code, equitable partition,…
We study codes with parameters of $q$-ary shortened Hamming codes, i.e., $(n=(q^m-q)/(q-1), q^{n-m}, 3)_q$. Firstly, we prove the fact mentioned in 1998 by Brouwer et al. that such codes are optimal, generalizing it to a bound for multifold…
We construct new families of completely regular codes by concatenation methods. By combining parity check matrices of cyclic Hamming codes, we obtain families of completely regular codes. In all cases, we compute the intersection array of…
Extended $1$-perfect codes in the Hamming scheme $H(n,q)$ can be equivalently defined as codes that turn to $1$-perfect codes after puncturing in any coordinate, as completely regular codes with certain intersection array, as uniformly…
A known Kronecker construction of completely regular codes has been investigated taking different alphabets in the component codes. This approach is also connected with lifting constructions of completely regular codes. We obtain several…
This work is a survey on completely regular codes. Known properties, relations with other combinatorial structures and constructions are stated. The existence problem is also discussed and known results for some particular cases are…