Related papers: Small Strong Blocking Sets by Concatenation
A family of random matrices is said to converge strongly to a limiting family of operators if the operator norm of every noncommutative polynomial of the matrices converges to that of the limiting operators. Recent developments surrounding…
For a graph $G$ in which vertices are either black or white, a zero forcing process is an iterative vertex color changing process such that the only white neighbor of a black vertex becomes black in the next time step. A zero forcing set is…
There are several methods for constructing secret sharing schemes, one of which is based on coding theory. Theoretically, every linear code can be used to construct secret sharing schemes. However, in general, determining the access…
Low-density parity-check (LDPC) convolutional codes have been shown to exhibit excellent performance under low-complexity belief-propagation decoding [1], [2]. This phenomenon is now termed threshold saturation via spatial coupling. The…
The detection of weak and rare effects in large amounts of data arises in a number of modern data analysis problems. Known results show that in this situation the potential of statistical inference is severely limited by the large-scale…
We propose a systematic design of protograph-based quasi-cyclic (QC) low-density parity-check (LDPC) codes with low error floor. We first characterize the trapping sets of such codes and demonstrate that the QC structure of the code…
We study the smallest possible number of points in a topological space having k open sets. Equivalently, this is the smallest possible number of elements in a poset having k order ideals. Using efficient algorithms for constructing a…
Building on previous results of Xing, we give new lower bounds on the rate of intersecting codes over large alphabets. The proof is constructive, and uses algebraic geometry, although nothing beyond the basic theory of linear systems on…
We deal with the random combinatorial structures called assemblies. By weakening the logarithmic condition which assures regularity of the number of components of a given order, we extend the notion of logarithmic assemblies. Using the…
In this paper, we study minimal (with respect to inclusion) zero forcing sets. We first investigate when a graph can have polynomially or exponentially many distinct minimal zero forcing sets. We also study the maximum size of a minimal…
Locating-dominating codes in a graph find their application in sensor networks and have been studied extensively over the years. A locating-dominating code can locate one object in a sensor network, but if there is more than one object, it…
The (constrained) minimization of a ratio of set functions is a problem frequently occurring in clustering and community detection. As these optimization problems are typically NP-hard, one uses convex or spectral relaxations in practice.…
The puncturing and shortening technique are two important approaches to constructing new linear codes from old ones. In the past 70 years, a lot of progress on the puncturing technique has been made, and many works on punctured linear codes…
The homological scaffold leverages persistent homology to construct a topologically sound summary of a weighted network. However, its crucial dependency on the choice of representative cycles hinders the ability to trace back global…
In this paper we introduce and investigate rank-metric intersecting codes, a new class of linear codes in the rank-metric context, inspired by the well-studied notion of intersecting codes in the Hamming metric. A rank-metric code is said…
Reduction of combinatorial filters involves compressing state representations that robots use. Such optimization arises in automating the construction of minimalist robots. But exact combinatorial filter reduction is an NP-complete problem…
The task of identifying resolving sets has been extensively studied due to its wide relevance in fields such as chemistry, robot navigation, combinatorial optimization, pattern recognition, and image processing. These applications have…
Batch codes serve as critical tools for load balancing in distributed storage systems. While numerous constructions exist for specific batch sizes t, current methodologies predominantly rely on code dimension parameters, limiting their…
In this paper, we investigate absorbing sets, responsible of error floors in Low Density Parity Check codes. We look for a concise, quantitative way to rate the absorbing sets' dangerousness. Based on a simplified model for iterative…
We prove that approximating the size of stopping and trapping sets in Tanner graphs of linear block codes, and more restrictively, the class of low-density parity-check (LDPC) codes, is NP-hard. The ramifications of our findings are that…