Related papers: Generalized Index Coding Problem and Discrete Poly…
Index coding, or broadcasting with side information, is a network coding problem of most fundamental importance. In this problem, given a directed graph, each vertex represents a user with a need of information, and the neighborhood of each…
Until this work, the packing radius of a poset code was only known in the cases where the poset was a chain, a hierarchy, a union of disjoint chains of the same size, and for some families of codes. Our objective is to approach the general…
We introduce a technique for establishing and amplifying gaps between parameters of network coding and index coding. The technique uses linear programs to establish separations between combinatorial and coding-theoretic parameters and…
Motivated by an application to database linear querying, such as private information-retrieval protocols, we suggest a fundamental property of linear codes -- the generalized covering radius. The generalized covering-radius hierarchy of a…
Combinatorial optimization is a fundamental problem found in many fields. In many real life situations, the constraints and the objective function forming the optimization problem are naturally distributed amongst different sites in some…
This paper contributes to the study of rank-metric codes from an algebraic and combinatorial point of view. We introduce $q$-polymatroids, the $q$-analogue of polymatroids, and develop their basic properties. We associate a pair of…
The two-sender unicast index coding problem is the most fundamental multi-sender index coding problem. The two senders collectively cater to the demands of all the receivers, by taking advantage of the knowledge of their side-information.…
We investigate group coding for arbitrary finite groups acting linearly on a vector space. These yield robust codes based on real or complex matrix groups. We give necessary and sufficient conditions for correct subgroup decoding using…
In the pliable variant of index coding, receivers are allowed to decode any new message not known a priori. Optimal code design for this variant involves identifying each receiver's choice of a new message that minimises the overall…
This dissertation presents new results on three different themes all related to matroid polytopes. First we investigate properties of Ehrhart polynomials of matroid polytopes, independence matroid polytopes, and polymatroids. We prove that…
Matroids, particularly linear ones, have been a powerful tool in parameterized complexity for algorithms and kernelization. They have sped up or replaced dynamic programming. Delta-matroids generalize matroids by encapsulating structures…
It is well known that linear rank-metric codes give rise to q-polymatroids. Analogously to matroid theory one may ask whether a given q-polymatroid is representable by a rank-metric code. We provide an answer by presenting an example of a…
Motivated by cryptographic applications such as predicate encryption, we consider the problem of representing an arbitrary predicate as the inner product predicate on two vectors. Concretely, fix a Boolean function $P$ and some modulus $q$.…
Generalized concatenated codes were introduced in the 1970s by Zinoviev. There are many types of codes in the literature that are known by other names that can be viewed as generalized concatenated codes. Examples include matrix-product…
This paper introduces combinatorial representations, which generalise the notion of linear representations of matroids. We show that any family of subsets of the same cardinality has a combinatorial representation via matrices. We then…
We give new characterizations for the class of uniformly dense matroids and study applications of these characterizations to graphic and real representable matroids. We show that a matroid is uniformly dense if and only if its base polytope…
In the recent years, several polynomial algorithms of a dynamical nature have been proposed to address the graph isomorphism problem. In this paper we propose a generalization of an approach exposed in cond-mat/0209112 and find that this…
In this paper we consider the problem of minimizing a general quadratic function over the mixed integer points in an ellipsoid. This problem is strongly NP-hard, NP-hard to approximate within a constant factor, and optimal solutions can be…
Index coding, a source coding problem over broadcast channels, has been a subject of both theoretical and practical interest since its introduction (by Birk and Kol, 1998). In short, the problem can be defined as follows: there is an input…
This paper establishes several new facts on generalized polyhedral convex sets and shows how they can be used in vector optimization. Among other things, a scalarization formula for the efficient solution sets of generalized vector…