Related papers: Bounds for flag codes
We study asymptotic lower and upper bounds for the sizes of constant dimension codes with respect to the subspace or injection distance, which is used in random linear network coding. In this context we review known upper bounds and show…
We consider a network coding setting where some of the messages and edges have fixed alphabet sizes, that do not change when we increase the common alphabet size of the rest of the messages and edges. We prove that the problem of deciding…
A large family of linear codes with flexible parameters from almost bent functions and perfect nonlinear functions are constructed and their parameters are determined. Some constructed linear codes and their related codes are optimal in the…
We consider the problem of error control in a coded, multicast network, focusing on the scenario where the errors can occur only on a proper subset of the network edges. We model this problem via an adversarial noise, presenting a formal…
For nonnegative integers $n$ and $d$, let $A(n,d)$ be the maximum cardinality of a binary code of length $n$ and minimum distance at least $d$. We consider a slight sharpening of the semidefinite programming bound of Gijswijt, Mittelmann…
In recent years, network coding has emerged as an innovative method that helps wireless network approaches its maximum capacity, by combining multiple unicasts in one broadcast. However, the majority of research conducted in this area is…
Motivated by applications in DNA-based data storage, constrained codes have attracted a considerable amount of attention from both academia and industry. We study the maximum cardinality of constrained codes for which the constraints can be…
For a given number of colors, $s$, the guessing number of a graph is the (base $s$) logarithm of the cardinality of the largest family of colorings of the vertex set of the graph such that the color of each vertex can be determined from the…
We investigate the maximum cardinality and the mathematical structure of error-correcting codes endowed with the Kendall-$\tau$ metric. We establish an averaging bound for the cardinality of a code with prescribed minimum distance, discuss…
Computing the crossing number of a graph is one of the most classical problems in computational geometry. Both it and numerous variations of the problem have been studied, and overcoming their frequent computational difficulty is an active…
Coding schemes with extremely low computational complexity are required for particular applications, such as wireless body area networks, in which case both very high data accuracy and very low power-consumption are required features. In…
This paper considers a multi-source multi-relay network, in which relay nodes employ a coding scheme based on random linear network coding on source packets and generate coded packets. If a destination node collects enough coded packets, it…
This paper investigates the problem of single-source multicasting over a communication network in the presence of restricted adversaries. When the adversary is constrained to operate only on a prescribed subset of edges, classical cut-set…
The study of codes for powerlines communication has garnered much interest over the past decade. Various types of codes such as permutation codes, frequency permutation arrays, and constant composition codes have been proposed over the…
We study an analogue of the Ramsey multiplicity problem for additive structures, in particular establishing the minimum number of monochromatic 3-APs in 3-colorings of $\mathbb{F}_3^n$ as well as obtaining the first non-trivial lower bound…
Let $\Gamma$ denote a distance-regular graph. The maximum size of codewords with minimum distance at least $d$ is denoted by $A(\Gamma,d)$. Let $\square_n$ denote the folded $n$-cube $H(n,2)$. We give an upper bound on $A(\square_n,d)$…
We consider the problem of minimizing the number of broadcasts for collecting all sensor measurements at a sink node in a noisy broadcast sensor network. Focusing first on arbitrary network topologies, we provide (i) fundamental limits on…
Given a finite field F_q and a positive integer n, a flag is a sequence of nested F_q-subspaces of a vector space F_q^n and a flag code is a nonempty collection of flags. The projected codes of a flag code are the constant dimension codes…
The Slepian-Wolf bound on the admissible coding rate forms the most fundamental aspect of distributed source coding. As such, it is necessary to provide a framework with which to model more practical scenarios with respect to the…
Upper and lower bounds are derived for the quantity in the title, which is tabulated for modest values of $n$ and $k.$ An application to graphs with many cycles is given.