Related papers: On a class of twisted elliptic curve codes
We study high-dimensional random geometric graphs (RGGs) of edge-density $p$ with vertices uniformly distributed on the $d$-dimensional torus and edges inserted between sufficiently close vertices with respect to an $L_q$-norm. We focus on…
For an elliptic curve $E/\Q$, we determine the maximum number of twists $E^d/\Q$ it can have such that $E^d(\Q)_{tors}\supsetneq E(\Q)[2]$. We use these results to determine the number of distinct quadratic fields $K$ such that…
Let K be a field of characteristic different from 2 and C an elliptic curve over K given by a Weierstrass equation. To divide an element of the group C by 2, one must solve a certain quartic equation. We characterise the quartics arising…
We investigate variations of Selmer ranks under quadratic twists satisfying the Heegner hypothesis. In particular, starting with an elliptic curve $E/\mathbb{Q}$ with partial $2$-torsion and a common relaxed Selmer group, we derive explicit…
PhD thesis investigating homological quantum codes derived from curved and higher dimensional geometries. In the first part we will consider closed surfaces with constant negative curvature. We show how such surfaces can be constructed and…
CSS codes are in one-to-one correspondance with length 3 chain complexes. The latter are naturally endowed with a tensor product $\otimes$ which induces a similar operation on the former. We investigate this operation, and in particular its…
In this paper, we study the existence of twisted constant scalar curvature K\"{a}hler (cscK) metrics and non-existence of coupled cscK metrics on minimal ruled surfaces over a Riemann surface of genus $2$. Moreover, we give a bound for the…
We study twisted derived equivalences for schemes in the setting of spectral algebraic geometry. To this end, we introduce the notion of a twisted equivalence and show that a twisted equivalence for perfect spectral algebraic stacks…
In tropical geometry, given a curve in a toric variety, one defines a corresponding graph embedded in Euclidean space. We study the problem of reversing this process for curves of genus zero and one. Our methods focus on describing curves…
We study the functional codes $C_2(X)$ defined on projective varieties $X$, in the case where $X\subset \mathbb{P}^3$ is a 1-degenerate quadric or a non-degenerate quadric (hyperbolic or elliptic). We find the minimum distance of these…
CC-GLPDC codes are a class of generalized low-density parity-check (GLDPC) codes where the constraint nodes (CNs) represent convolutional codes. This allows for efficient decoding in the trellis with the forward-backward algorithm, and the…
We construct infinite families of pairs of (geometrically non-isogenous) elliptic curves defined over $\mathbb{Q}$ with $12$-torsion subgroups that are isomorphic as Galois modules. This extends previous work of Chen and Fisher where it is…
The constraint complexity of a graphical realization of a linear code is the maximum dimension of the local constraint codes in the realization. The treewidth of a linear code is the least constraint complexity of any of its cycle-free…
We show that the sequence of dimensions of the linear spaces, generated by a given rank-metric code together with itself under several applications of a field automorphism, is an invariant for the whole equivalence class of the code. The…
We introduce a unified ensemble for turbo-like codes (TCs) that contains the four main classes of TCs: parallel concatenated codes, serially concatenated codes, hybrid concatenated codes, and braided convolutional codes. We show that for…
Twisted hypercubes are generalizations of the Boolean hypercube, obtained by iteratively connecting two instances of a graph by a uniformly random perfect matching. Dudek et al. showed that when the two instances are independent, these…
We study geometric properties of linear strata of uni-singular curves. The singularities of closures of the strata are resolved and the resolutions are represent as projective bundles. This enables to study their geometry. In particular we…
Maximum Distance Separable (MDS) self-dual codes are of significant theoretical and practical importance. Generalized Reed-Solomon (GRS) codes are the most prominent MDS codes. Correspondingly there have been many research on constructions…
On a space of stable maps, the psi classes are modified by subtracting certain boundary divisors. The top products of modified psi classes, usual psi classes, and classes pulled back along the evaluation maps are called twisted descendants;…
We provide an algorithm to check whether two rational space curves are related by a similarity. The algorithm exploits the relationship between the curvatures and torsions of two similar curves, which is formulated in a computer algebra…