Related papers: A Combinatorial Approach to Flag Codes
This paper investigates the construction of rank-metric codes with specified Ferrers diagram shapes. These codes play a role in the multilevel construction for subspace codes. A conjecture from 2009 provides an upper bound for the dimension…
We introduce generalized function-correcting partition codes (GFCPCs) that simultaneously protect multiple partitions of the message space against different numbers of errors. Given partitions with respective distance requirements, a GFCPC…
In various application fields, such as fluid-, cell-, or crowd-simulations, spatial data structures are very important. They answer nearest neighbor queries which are instrumental in performing necessary computations for, e.g., taking the…
We introduce \emph{hierarchical depth}, a new invariant of line bundles and divisors, defined via maximal chains of effective sub-line bundles. This notion gives rise to \emph{hierarchical filtrations}, refining the structure of the Picard…
We present a general theory to obtain linear network codes utilizing forms and obtain explicit families of equidimensional vector spaces, in which any pair of distinct vector spaces intersect in the same small dimension. The theory is…
An embedding of a code is a mapping that preserves distances between codewords. We prove that any code with code distance $\rho$ and length $d$ can be embedded into an MDS code with the same code distance and length but under a larger…
Coded distributed computing introduced by Li et al. in 2015 is an efficient approach to trade computing power to reduce the communication load in general distributed computing frameworks such as MapReduce. In particular, Li et al. show that…
Convex neural codes are subsets of the Boolean lattice that record the intersection patterns of convex sets in Euclidean space. Much work in recent years has focused on finding combinatorial criteria on codes that can be used to classify…
Persistence diagrams have been widely used to quantify the underlying features of filtered topological spaces in data visualization. In many applications, computing distances between diagrams is essential; however, computing these distances…
A textile structure is a periodic arrangement of threads in the thickened plane. A topological classification of textile structures is harder than for classical knots and links that are non-periodic and restricted to a bounded region. The…
Network coding is a technique to maximize communication rates within a network, in communication protocols for simultaneous multi-party transmission of information. Linear network codes are examples of such protocols in which the local…
In this article, we define and study a geometry and an order on the set of partitions of an even number of objects. One of the definitions involves the partition algebra, a structure of algebra on the set of such partitions depending on an…
Graphs are a natural representation for systems based on relations between connected entities. Combinatorial optimization problems, which arise when considering an objective function related to a process of interest on discrete structures,…
A detection system, modeled in a graph, is composed of "detectors" positioned at a subset of vertices in order to uniquely locate an ``intruder" at any vertex. \emph{Identifying codes} use detectors that can sense the presence or absence of…
We study the algorithmic aspect of edge bundling. A bundled crossing in a drawing of a graph is a group of crossings between two sets of parallel edges. The bundled crossing number is the minimum number of bundled crossings that group all…
The brain encodes spacial structure through a combinatorial code of neural activity. Experiments suggest such codes correspond to convex areas of the subject's environment. We present an intrinsic condition that implies a neural code may…
A new family of error-correcting codes, called Fourier codes, is introduced. The code parity-check matrix, dimension and an upper bound on its minimum distance are obtained from the eigenstructure of the Fourier number theoretic transform.…
Graph-structured data is central to many scientific and industrial domains, where the goal is often to optimize objectives defined over graph structures. Given the combinatorial complexity of graph spaces, such optimization problems are…
Graph code is a linear code obtained from linear codes $C$ and a certain bipartite graph G. In this paper, I propose an expansion of the definition of graph code to general $l$-partite, and give its lower bound of minimum distance. I also…
Fractional graph isomorphism is the linear relaxation of an integer programming formulation of graph isomorphism. It preserves some invariants of graphs, like degree sequences and equitable partitions, but it does not preserve others like…