Related papers: Weighted Coordinates Poset Block Codes
In this paper, we determine the complete weight distribution of the space $ \mathbb{F}_q^N $ endowed by the weighted coordinates poset block metric ($(P,w,\pi)$-metric), also known as the $(P,w,\pi)$-space, thereby obtaining it for…
Given $[n]=\{1,2,\ldots,n\}$, a poset order $\preceq$ on $[n]$, a label map $\pi : [n] \rightarrow \mathbb{N}$ defined by $\pi(i)=k_i$ with $\sum_{i=1}^{n}\pi (i) = N$, and a weight function $w$ on $\mathbb{F}_{q}$, let $\mathbb{F}_{q}^N$…
Weighted poset block metric is a generalization of weighted poset metric introduced by Panek et al. ([\ref{panek}]) and the metric for linear error-block codes introduced by Feng et al. ([\ref{FENG}]). This type of metrics includes many…
Given a regular multiset $M$ on $[n]=\{1,2,\ldots,n\}$, a partial order $R$ on $M$, and a label map $\pi : [n] \rightarrow \mathbb{N}$ defined by $\pi(i) = k_i$ with $\sum_{i=1}^{n}\pi (i) = N$, we define a pomset block metric…
In this paper, we introduce a new family of metrics, weighted poset block metric, that combine the weighted coordinates poset metric introduced by Panek et al. [(\ref{panek})] and the metric for linear error-block codes introduced by Feng…
In this paper, we introduce codes equipped with pomset block metric. A Singleton type bound for pomset block codes is obtained. Code achieving the Singleton bound, called a maximum distance separable code (for short, MDS…
In this paper, we establish the Singleton bound for pomset block codes ($(Pm,\pi)$-codes) of length $N$ over the ring $\mathbb{Z}_m$. We give a necessary condition for a code to be MDS in the pomset (block) metric and prove that every MDS…
Let $P$ be a partial order on $[n] = \{1,2,\ldots,n\}$, $\mathbb{F}_{q}^n$ be the linear space of $n$-tuples over a finite field $\mathbb{F}_{q}$ and $w$ be a weight on $\mathbb{F}_{q}$. In this paper, we consider metrics on…
We prove a simple formula for arbitrary cluster variables in the marked surfaces model. As part of the formula, we associate a labeled poset to each tagged arc, such that the associated $F$-polynomial is a weighted sum of order ideals. Each…
Given complex numbers $w_1, \ldots, w_n$, we define the weight $w(X)$ of a set $X$ of 0-1 vectors as the sum of $w_1^{x_1} \cdots w_n^{x_n}$ over all vectors $(x_1, \ldots, x_n)$ in $X$. We present an algorithm, which for a set $X$ defined…
Etzion et al. introduced metrics on $\mathbb{F}_2^n$ based on directed graphs on $n$ vertices and developed some basic coding theory on directed graph metric spaces. In this paper, we consider the problem of classifying directed graphs…
Let $P = (\{1,2,\ldots,n,\leq)$ be a poset that is an union of disjoint chains of the same length and $V=\mathbb{F}_q^N$ be the space of $N$-tuples over the finite field $\mathbb{F}_q$. Let $V_i = \mathbb{F}_q^{k_i}$, $1 \leq i \leq n$, be…
We consider the $[q+1,q-3,5]_q3$ generalized doubly-extended Reed-Solomon code of codimension $4$ as the code associated with the twisted cubic in the projective space $\mathrm{PG}(3,q)$. Basing on the point-plane incidence matrix of…
We consider a generalized poset sorting problem (GPS), in which we are given a query graph $G = (V, E)$ and an unknown poset $\mathcal{P}(V, \prec)$ that is defined on the same vertex set $V$, and the goal is to make as few queries as…
We study $q$-ary codes with distance defined by a partial order of the coordinates of the codewords. Maximum Distance Separable (MDS) codes in the poset metric have been studied in a number of earlier works. We consider codes that are close…
The weight distribution of the cosets of maximum distance separable (MDS) codes is considered. In 1990, P.G. Bonneau proposed a relation to obtain the full weight distribution of a coset of an MDS code with minimum distance $d$ using the…
In this paper, we define the weighted homogeneous space (WHS), denoted by $\frac{G}{P}[\psi_H]$ where $\psi_H$ is weight function defined on the set of simple roots of $G$, by an element $H$ in the highest Weyl chamber. The weight function…
The vertex PI index is a distance--based molecular structure descriptor, that recently found numerous chemical applications. In order to increase diversity of this topological index for bipartite graphs, we introduce weighted version…
We consider the poset of weighted partitions $\Pi_n^w$, introduced by Dotsenko and Khoroshkin in their study of a certain pair of dual operads. The maximal intervals of $\Pi_n^w$ provide a generalization of the lattice $\Pi_n$ of…
In this paper we extend the concept of generalized Wei weights for poset-weight codes and show that all linear codes C satisfy the chain condition if support of C is a subposet totally ordered.