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We define linear and semilinear isometry for general subspace codes, used for random network coding. Furthermore, some results on isometry classes and automorphism groups of known constant dimension code constructions are derived.
Given a ternary homogeneous polynomial, the fixed points of the map from $\mathbb{P}^2$ to itself defined by its gradient are called its eigenpoints. We focus on cubic polynomials, and analyze configurations of eigenpoints that admit one or…
The purpose of this paper is to present the structure of the linear codes over a finite field with q elements that have a permutation automorphism of order m. These codes can be considered as generalized quasi-cyclic codes. Quasi-cyclic…
We analyze the nonlinear optics of quasi one-dimensional quantum graphs and manipulate their topology and geometry to generate for the first time nonlinearities in a simple system approaching the fundamental limits of the first and second…
Two isometry groups of combinatorial codes are described: the group of automorphisms and the group of monomial automorphisms, which is the group of those automorphisms that extend to monomial maps. Unlike the case of classical linear codes,…
We consider spaces of plane curves in the setting of algebraic geometry and of singularity theory. On one hand there are the complete linear systems, on the other we consider unfolding spaces of bivariate polynomials of Brieskorn-Pham type.…
The aim of this paper is to study symmetries of linearly singular differential equations, namely, equations that can not be written in normal form because the derivatives are multiplied by a singular linear operator. The concept of…
A lot of attention has been paid to the investigation of the algebraic properties of linear codes. In most cases, this investigation involves the determination of required code automorphisms, which are useful for decoders, such as the…
Many physical systems can be described by nonlinear eigenvalues and bifurcation problems with a linear part that is non-selfadjoint e.g. due to the presence of loss and gain. The balance of these effects is reflected in an antilinear…
A class of differential calculi is explored which is determined by a set of automorphisms of the underlying associative algebra. Several examples are presented. In particular, differential calculi on the quantum plane, the $h$-deformed…
We investigate geometric properties of surfaces given by certain formulae. In particular, we calculate the singular curvature and the limiting normal curvature of such surfaces along the set of singular points consisting of singular points…
We investigate (local) automorphisms of parabolic geometries that generalize geodesic symmetries. We show that many types of parabolic geometries admit at most one generalized geodesic symmetry at a point with non-zero harmonic curvature.…
If the bimodule of 1-forms of a differential calculus over an associative algebra is the direct sum of 1-dimensional bimodules, a relation with automorphisms of the algebra shows up. This happens for some familiar quantum space calculi.
In this paper we study the algebraic geometry of any two-point code on the Hermitian curve and reveal the purely geometric nature of their dual minimum distance. We describe the minimum-weight codewords of many of their dual codes through…
In this article we study the automorphism groups of binary cyclic codes. In particular, we provide explicit constructions for codes whose automorphism groups can be described as (a) direct products of two symmetric groups or (b) iterated…
The main purpose of this paper is to describe some published results and outline corresponding approaches which when applied to automorphism groups of algebras or groups establish that these groups are linear or non-linear.
We study linearizability of actions of finite groups on cubic threefolds with non-isolated singularities.
A method of constructing algebraic-geometric codes with many automorphisms arising from Galois points for algebraic curves is presented.
There are several notions of duality between lines and points. In this note, it is shown that all these can be studied in a unified way. Most interesting properties are independent of specific choices. It is also shown that either dual…
In this report we consider the parameterization of low-dimensional manifolds that are specified (approximately) by a set of points very close to the manifold in the original high-dimensional space. Our objective is to obtain a…