Related papers: Multi-Point AG Codes on the GK Maximal Curves
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,…
Let N_q(g) denote the maximal number of F_q-rational points on any curve of genus g over the finite field F_q. Ihara (for square q) and Serre (for general q) proved that limsup_{g-->infinity} N_q(g)/g > 0 for any fixed q. In their proofs…
In this paper we initiate the study of cyclic algebraic geometry codes. We give conditions to construct cyclic algebraic geometry codes in the context of algebraic function fields over a finite field by using their group of automorphisms.…
Algebraic-geometric codes on Garcia-Stichtenoth family of curves are used to construct the asymptotically good quantum codes.
We introduce an algebraicity criteria. It has the following form: under certain conditions, an analytic subvariety of some algebriac variety over a global field $K$, if it contains many $K$-points, then it is algebraic over $K.$ This gives…
We determine the quantum automorphism groups of finite graphs. These are quantum subgroups of the quantum permutation groups defined by Wang. The quantum automorphism group is a stronger invariant for finite graphs than the usual one. We…
We prove a tropical analogue of the theorem of Hurwitz: a leafless metric graph of genus $g \ge 2$ has at most $12$ automorphisms when $g = 2$; $2^g g!$ automorphisms when $g \ge 3$. These inequalities are optimal; for each genus, we give…
We give a combinatorial description of closed curves on oriented surfaces in terms of certain permutations, called charts. We describe automorphisms of curves in terms of charts and compute the total number of curves counted with…
These notes concern aspects of various graphs whose vertex set is a group $G$ and whose edges reflect group structure in some way (so that they are invariant under the action of the automorphism group of $G$). The graphs I will discuss are…
We study hyperelliptic curves C with an action of an affine group of automorphisms G. We establish a closed form expression for the quotient curve C/G and for the first etale cohomology group of C as a representation of G. The motivation…
Cyclic AG-codes $C_L(D,G)$ on the Hermitian curve $H_q$ over $\mathbb{F}_{q^2}$ are constructed such that $G = m(P_2 + \ldots + P_q)$, where $2 \le m \le q-1$ and $\mathrm{supp}(G)$ is the intersection of $H_q$ with a chord $\ell$ minus two…
We present a list of all polynomial dominanting maps of the complex plane with branched value curve isomorphic to the complex line, up to polynomial automorphisms.
We consider the infinite-dimensional hypercube graph. This graph is not connected and has isomorphic connected components. We describe the restrictions of its automorphisms to the connected components and the automorphism group of connected…
It is proved that for every $d\ge 2$ such that $d-1$ divides $q-1$, where $q$ is a power of 2, there exists a Denniston maximal arc $A$ of degree $d$ in $\PG(2,q)$, being invariant under a cyclic linear group that fixes one point of $A$ and…
We apply a reduction to the Beauville systems to obtain a family of new algebraic completely integrable systems, related to curves with a cyclic automorphism.
Some linear codes associated to maximal algebraic curves via Feng-Rao construction are investigated. In several case, these codes have better minimum distance with respect to the previously known linear codes with same length and dimension.
We give a method of generating strongly polynomial sequences of graphs, i.e., sequences $(H_{\mathbf{k}})$ indexed by a multivariate parameter $\mathbf{k}=(k_1,\ldots, k_h)$ such that, for each fixed graph $G$, there is a multivariate…
We study arithmetical and geometrical properties of {\it maximal curves}, that is, curves defined over the finite field $\mathbb F_{q^2}$ whose number of $\mathbb F_{q^2}$-rational points reachs the Hasse-Weil upper bound. Under a…
We construct group-invariant CR maps from the unit sphere in $\mathbb{C}^3$ and provide sharp bounds for the gap termination in this setting.
Two mappings in a finite field, the Frobenius mapping and the cyclic shift mapping, are applied on lines in PG($n,p$) or codes in the Grassmannian, to form automorphisms groups in the Grassmanian and in its codes. These automorphisms are…