Related papers: On Codes over $\mathbb{Z}_{p^2}$ and its Covering …
In this paper, we show that LCD codes are not equivalent to linear codes over small finite fields. The enumeration of binary optimal LCD codes is obtained. We also get the exact value of LD$(n,2)$ over $\mathbb{F}_3$ and $\mathbb{F}_4$. We…
In this paper, we defined the $Z_q$-linear codes and discussed its various parameters. We constructed $Z_q$-Simplex code and $Z_q$-MacDonald code and found its parameters. We have given a lower and an upper bounds of its covering radius for…
It's well known that the quadratic residue code over finite fields is an interesting class of cyclic codes for its higher minimum distance. Let $g$ be a positive integer and $p,p_{1},\ldots, p_{g}$ be distinct odd primes, the present paper…
In 1981, Schatz proved that the covering radius of the binary Reed-Muller code $RM(2,6)$ is 18. For $RM(2,7)$, we only know that its covering radius is between 40 and 44. In this paper, we prove that the covering radius of the binary…
A code $C$ is called $\mathbb{Z}_p\mathbb{Z}_{p^2}$-linear if it is the Gray image of a $\mathbb{Z}_p\mathbb{Z}_{p^2}$-additive code. For any prime number $p$ larger than $3$, the bounds of the rank of $\mathbb{Z}_p\mathbb{Z}_{p^2}$-linear…
A locally recoverable code (LRC code) is a code over a finite alphabet such that every symbol in the encoding is a function of a small number of other symbols that form a recovering set. Bounds on the rate and distance of such codes have…
We study the covering radii of $2$-transitive permutation groups of Lie rank one, giving bounds and links to finite geometry.
We study the covering path problem on a grid of R^{2}. We generalize earlier results on a rectangular grid and prove that the covering path cost can be bounded by the area and perimeter of the grid. We provide (2+\epsilon) and…
The more exact upper estimate of the percolation threshold for the {\it site problem} on the quadratic lattice ${\Bbb Z}^2$ have been found on the basis of the cluster decomposition. It is done by the number estimate of cycles on ${\Bbb…
Let $p$ be a prime number and $\varsigma$ and $m$ be a positive integers. Let $\mathcal{R} = \mathbb{F}_{2^m} + u\mathbb{F}_{2^m} + u^2\mathbb{F}_{2^m}$ ($u^3 = 0$). Cyclic codes of length $2^\varsigma$ over $\mathcal{R}$ are precisely the…
We give a complete classification of self-dual completely regular codes with covering radius $\rho \leq 3$. For $\rho=1$ the results are almost trivial. For $\rho=2$, by using properties of the more general class of uniformly packed codes…
We prove bounds for the covering numbers of classes of convex functions and convex sets in Euclidean space. Previous results require the underlying convex functions or sets to be uniformly bounded. We relax this assumption and replace it…
We investigate the $a$-numbers of $\mathbb{Z}/p^2\mathbb{Z}$-covers in characteristic $p>2$ and extend a technique originally introduced by Farnell and Pries for $\mathbb{Z}/p\mathbb{Z}$-covers. As an application of our approach, we…
In this note, we study the classification of $\mathbb{Z}_4$-codes. For some special cases $(k_1,k_2)$, by hand, we give a classification of $\mathbb{Z}_4$-codes of length $n$ and type $4^{k_1}2^{k_2}$ satisfying a certain condition. Our…
In graph theory, a tree is one of the more popular families of graphs with a wide range of applications in computer science as well as many other related fields. While there are several distance measures over the set of all trees, we…
In this paper, we introduce a new parameter of a code, referred to as the remoteness, which can be viewed as a dual to the covering radius. Indeed, the remoteness is the minimum radius needed for a single ball to cover all codewords. After…
We present a bound on the size of linear codes. This bound is independent of other known bounds, e.g. the Griesmer bound.
In this paper we present an extension of known semidefinite and linear programming upper bounds for spherical codes and consider a version of this bound for distance graphs. We apply the main result for the distance distribution of a…
We study the Singleton-type bound that provides an upper limit on the minimum distance of locally repairable codes. We present an improved bound by carefully analyzing the combinatorial structure of the repair sets. Thus, we show the…
We construct two series of linear codes over finite ring $\mathbb{F}_{q}[x]/(x^2)$ and Galois ring $GR(p^2,m)$ respectively reaching the Griesmer bound. They derive two series of codes over finite field $\mathbb{F}_{q}$ by Gray map. The…