Related papers: Almost perfect linear Lee codes of packing radius …
Linear complementary dual (LCD) codes over finite fields are linear codes satisfying $C\cap C^{\perp}=\{0\}$. We generalize the LCD codes over finite fields to $\mathbb{Z}_2\mathbb{Z}_2[u]$-LCD codes over the ring…
For an $m$-cycle $C$, an inside $m$-cycle of $C$ is a cycle on the same vertex set, that is edge-disjoint from $C$. In an $m$-cycle system, $(\mathcal{X}, \mathcal{C})$, if inside $m$-cycles can be chosen -one for each cycle- to form…
Linear complementary dual (LCD) codes can be used to against side-channel attacks and fault noninvasive attacks. Let $d_{a}(n,6)$ and $d_{l}(n,6)$ be the minimum weights of all binary optimal linear codes and LCD codes with length $n$ and…
We consider $t$-Lee-error-correcting codes of length $n$ over the residue ring $\mathbb{Z}_m := \mathbb{Z}/m\mathbb{Z}$ and determine upper and lower bounds on the number of $t$-Lee-error-correcting codes. We use two different methods,…
Nearly orthogonal lattices were formally defined in [4], where their applications to image compression were also discussed. The idea of ``near orthogonality" in $2$-dimensions goes back to the work of Gauss. In this paper, we focus on…
An important question in the study of quasi-perfect codes is whether such codes can be constructed for all possible lengths $n$. In this paper, we address this question for specific values of $n$. First, we investigate the existence of…
This paper provides a link between matroid theory and locally repairable codes (LRCs) that are either linear or more generally almost affine. Using this link, new results on both LRCs and matroid theory are derived. The parameters…
The existence question for tiling of the $n$-dimensional Euclidian space by crosses is well known. A few existence and nonexistence results are known in the literature. Of special interest are tilings of the Euclidian space by crosses with…
The aim of this work is a systematic investigation of the possible parameters of quasi-perfect (QP) binary and ternary linear codes of small dimensions and preparing a complete classification of all such codes. First we give a list of…
MDS codes have diverse practical applications in communication systems, data storage, and quantum codes due to their algebraic properties and optimal error-correcting capability. In this paper, we focus on a class of linear codes and…
We study $1$-perfect codes in Doob graphs $D(m,n)$. We show that such codes that are linear over $GR(4^2)$ exist if and only if $n=(4^{g+d}-1)/3$ and $m=(4^{g+2d}-4^{g+d})/6$ for some integers $g \ge 0$ and $d>0$. We also prove necessary…
Perfect error correcting codes allow for an optimal transmission of information while guaranteeing error correction. For this reason, proving their existence has been a classical problem in both pure mathematics and information theory.…
Recently, linear codes constructed from defining sets have been studied extensively. They may have nice parameters if the defining set is chosen properly. Let $ m >2$ be a positive integer. For an odd prime $ p $, let $ r=p^m $ and…
Linear complementary dual codes (or codes with complementary duals) are codes whose intersections with their dual codes are trivial. We study binary linear complementary dual $[n,k]$ codes with the largest minimum weight among all binary…
We describe odd-length-cube tilings of the n-dimensional q-ary torus what includes q-periodic integer lattice tilings of R^n. In the language of coding theory these tilings correspond to perfect codes with respect to the maximum metric. A…
We prove that the class of $\Z_2\Z_2[u]$-linear codes is exactly the class of $\Z_2$-linear codes with automorphism group of even order. Using this characterization, we give examples of known codes, e.g. perfect codes, which has a…
Perfect codes in the $n$-dimensio\-nal grid $\Lambda_n$ 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…
The Fibonacci cube of dimension n, denoted as $\Gamma$ n , is the subgraph of the n-cube 5 Q n induced by vertices with no consecutive 1's. Ashrafi and his co-authors proved the non-existence of perfect codes in $\Gamma$ n for n $\ge$ 4. As…
A binary poset code of codimension M (of cardinality 2^{N-M}, where N is the code length) can correct maximum M errors. All possible poset metrics that allow codes of codimension M to be M-, (M-1)- or (M-2)-perfect are described. Some…
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…