Related papers: Intersecting Codes and the Connectivity of $q$-Mat…
The problem of error control in random linear network coding is addressed from a matrix perspective that is closely related to the subspace perspective of K\"otter and Kschischang. A large class of constant-dimension subspace codes is…
It is shown that the Whitney function of a representable q-matroid and the collection of all higher weight enumerators of any representing rank-metric code determine each other via a monomial substitution. Moreover, the q-matroid itself and…
Over a finite field $\mathbb{F}_{q^m}$, the evaluation of skew polynomials is intimately related to the evaluation of linearized polynomials. This connection allows one to relate the concept of polynomial independence defined for skew…
In this paper, we describe the properties of the $i$-components of Hamming codes. We suggest constructions of the admissible families of components of Hamming codes. It is shown that every $q$-ary code of length $m$ and minimum distance 5…
We propose a family of surface codes with general lattice structures, where the error-tolerances against bit and phase errors can be controlled asymmetrically by changing the underlying lattice geometries. The surface codes on various…
We show how the theory of affine geometries over the ring ${\mathbb Z}/\langle q - 1\rangle$ can be used to understand the properties of toric and generalized toric codes over ${\mathbb F}_q$. The minimum distance of these codes is strongly…
In this paper, we investigate geometrical properties of the rank metric space and covering properties of rank metric codes. We first establish an analytical expression for the intersection of two balls with rank radii, and then derive an…
In this paper, we consider the tractability of the matroid intersection problem under the minimum rank oracle. In this model, we are given an oracle that takes as its input a set of elements and returns as its output the minimum of the…
Rank-metric codes, defined as sets of matrices over a finite field with the rank distance, have gained significant attention due to their applications in network coding and connections to diverse mathematical areas. Initially studied by…
Linear error-correcting codes can be used for constructing secret sharing schemes; however finding in general the access structures of these secret sharing schemes and, in particular, determining efficient access structures is difficult.…
Recent research on distributed storage systems (DSSs) has revealed interesting connections between matroid theory and locally repairable codes (LRCs). The goal of this chapter is to introduce the reader to matroids and polymatroids, and…
We address a specific case of the matroid intersection problem: given a set of graphs sharing the same set of vertices, select a minimum cycle basis for each graph to maximize the size of their intersection. We provide a comprehensive…
As part of the recent developments in infinite matroid theory, there have been a number of conjectures about how standard theorems of finite matroid theory might extend to the infinite setting. These include base packing, base covering, and…
We consider network coding for networks experiencing worst-case bit-flip errors, and argue that this is a reasonable model for highly dynamic wireless network transmissions. We demonstrate that in this setup prior network error-correcting…
We examine an error-correcting coding framework in which each coded symbol is constrained to be a function of a fixed subset of the message symbols. With an eye toward distributed storage applications, we seek to design systematic codes…
For a real affine hyperplane arrangement, we define an integer intersection matrix with a natural $q$-deformation related to the intersections of bounded chambers of the arrangement. By connecting the integer matrix to a bilinear form of…
This paper presents encoding and decoding algorithms for several families of optimal rank metric codes whose codes are in restricted forms of symmetric, alternating and Hermitian matrices. First, we show the evaluation encoding is the right…
A set function can be extended to the unit cube in various ways; the correlation gap measures the ratio between two natural extensions. This quantity has been identified as the performance guarantee in a range of approximation algorithms…
The intersection of a linear code with its dual is called the hull of the code. It is known that, for classical linear codes under the Hamming-metric, the dimension of the hull can be reduced up to equivalence. This phenomenon leads to the…
Designing quantum error correcting codes that promise a high error threshold, low resource overhead and efficient decoding algorithms is crucial to achieve large-scale fault-tolerant quantum computation. The concatenated quantum Hamming…