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The minimum distance is one of the most important combinatorial characterizations of a code. The maximum likelihood decoding problem is one of the most important algorithmic problems of a code. While these problems are known to be hard for…
In this Master of Science Thesis I introduce geometric algebra both from the traditional geometric setting of vector spaces, and also from a more combinatorial view which simplifies common relations and operations. This view enables us to…
The present paper suggests a new approach for geometric representation of 3D spatial models and provides a new compression algorithm for 3D meshes, which is based on mathematical theory of convex geometry. In our approach we represent a 3D…
We give a complete characterization of simple graphs whose adjacency matrices generate binary linear complementary dual (LCD) codes. In particular, we completely characterize a distance-regular graph which yields an LCD code in terms of the…
The persistence diagram (PD) is an increasingly popular topological descriptor. By encoding the size and prominence of topological features at varying scales, the PD provides important geometric and topological information about a space.…
We construct linear codes over the finite field Fq from arbitrary simplicial complexes, establishing a connection between topological properties and fundamental coding parameters. First, we study the behaviour of the weights of codewords…
A generic construction of linear codes over finite fields has recently received a lot of attention, and many one-weight, two-weight and three-weight codes with good error correcting capability have been produced with this generic approach.…
Braids can be represented geometrically as curve diagrams. The geometric complexity of a braid is the minimal complexity of a curve diagram representing it. We introduce and study the corresponding notion of geometric generating function.…
Geometric relational embeddings map relational data as geometric objects that combine vector information suitable for machine learning and structured/relational information for structured/relational reasoning, typically in low dimensions.…
Let $\mathcal{L}$ and $\mathcal{L}_0$ be the binary codes generated by the column $\mathbb{F}_2$-null space of the incidence matrix of external points versus passant lines and internal points versus secant lines with respect to a conic in…
Correspondence is a ubiquitous problem in computer vision and graph matching has been a natural way to formalize correspondence as an optimization problem. Recently, graph matching solvers have included higher-order terms representing…
Pseudo-geometric designs are combinatorial designs which share the same parameters as a finite geometry design, but which are not isomorphic to that design. As far as we know, many pseudo-geometric designs have been constructed by the…
The hull of a linear code $C$ is the intersection of $C$ with its dual code. We present and analyze the number of linear $q$-ary codes of the same length and dimension but with different dimensions for their hulls. We prove that for given…
The {\it linear representation} of a subset of a finite projective space is an incidence system of affine points and lines determined by the subset. In this paper we use character theory to show that the rank of the incidence matrix has a…
Let $\bar{\Gamma}$ be the point-hyperplane geometry of a projective space $\mathrm{PG(V)},$ where $V$ is a $(n+1)$-dimensional vector space over a finite field $\mathbb{F}_q$ of order $q.$ Suppose that $\sigma$ is an automorphism of…
In this paper, we explore a connection between binary hierarchical models, their marginal polytopes and codeword polytopes, the convex hulls of linear codes. The class of linear codes that are realizable by hierarchical models is…
Spine spaces can be considered as fragments of a projective Grassmann space. We prove that the structure of lines together with binary coplanarity relation, as well as with binary relation of being in one pencil of lines, is a sufficient…
Over fields of characteristic unequal to $2$, we can identify symmetric matrices with homogeneous polynomials of degree $2$. This allows us to view symmetric rank-metric codes as living inside the space of such polynomials. In this paper,…
Projective Reed-Muller codes correspond to subcodes of the Reed-Muller code in which the polynomials being evaluated to yield codewords, are restricted to be homogeneous. The Generalized Hamming Weights (GHW) of a code ${\cal C}$, identify…
We introduce an algorithm which can be directly used to feasible and optimum search in linear programming. Starting from an initial point the algorithm iteratively moves a point in a direction to resolve the violated constraints. At the…