Instituto de Investigación en Matemáticas (IMUVA)Instituto de Investigación en Matemáticas (IMUVA)https://uvadoc.uva.es/handle/10324/321972021-06-22T08:55:52Z2021-06-22T08:55:52ZEntanglement-assisted quantum error-correcting codes from RS codes and BCH codes with extension degree 2Galindo, CarlosHernando, FernandoRuano, Diegohttps://uvadoc.uva.es/handle/10324/464702021-05-21T22:23:41Z2021-01-01T00:00:00ZEntanglement-assisted quantum error correcting codes (EAQECCs) constructed from Reed-Solomon codes and BCH codes are considered in this work. It is provided a complete and explicit formula for the parameters of EAQECCs coming from any Reed-Solomon code, for the Hermitian metric, and from any BCH code with extension degree 2 and consecutive cyclotomic cosets, for both the Euclidean and the Hermitian metric. The main task in this work is the computation of a completely general formula for c, the minimum number of required maximally entangled quantum states.
2021-01-01T00:00:00ZHigh dimensional affine codes whose square has a designed minimum distanceGarcía Marco, IgnacioMárquez Corbella, IreneRuano Benito, Diegohttps://uvadoc.uva.es/handle/10324/459982021-05-21T22:23:40Z2020-01-01T00:00:00ZGiven a linear code C, its square code C(2) is the span of all component-wise products of two elements of C. Motivated by applications in multi-party computation, our purpose with this work is to answer the following question: which families of affine variety codes have simultaneously high dimension k(C) and high minimum distance of C(2), d(C(2))? More precisely, given a designed minimum distance d we compute an affine variety code C such that d(C(2))≥d and the dimension of C is high. The best constructions we propose mostly come from hyperbolic codes. Nevertheless, for small values of d, they come from weighted Reed–Muller codes
2020-01-01T00:00:00ZAsymmetric Entanglement-Assisted Quantum Error-Correcting Codes and BCH CodesGalindo, CarlosHernando, FernandoMatsumoto, RyutarohRuano Benito, Diegohttps://uvadoc.uva.es/handle/10324/459972021-05-21T22:23:40Z2020-01-01T00:00:00ZThe concept of asymmetric entanglement-assisted quantum error-correcting code (asymmetric EAQECC) is introduced in this article. Codes of this type take advantage of the asymmetry in quantum errors since phase-shift errors are more probable than qudit-flip errors. Moreover, they use pre-shared entanglement between encoder and decoder to simplify the theory of quantum error correction and increase the communication capacity. Thus, asymmetric EAQECCs can be constructed from any pair of classical linear codes over an arbitrary field. Their parameters are described and a Gilbert-Varshamov bound is presented. Explicit parameters of asymmetric EAQECCs from BCH codes are computed and examples exceeding the introduced Gilbert-Varshamov bound are shown.
2020-01-01T00:00:00ZForwards attraction properties in scalar non-autonomous linear-dissipative parabolic PDEs. The case of null upper Lyapunov exponentLanga, José AntonioObaya García, RafaelSanz Gil, Ana Maríahttps://uvadoc.uva.es/handle/10324/416032021-05-21T22:23:39Z2020-01-01T00:00:00ZAs it is well-known, the forwards and pullback dynamics are in general unrelated. In this paper we present an in-depth study of whether the pullback attractor is also a forwards attractor for the processes involved with the skew-product semiflow induced by a family of scalar non-autonomous reaction-diffusion equations which are linear in a neighbourhood of zero and have null upper Lyapunov exponent. Besides, the notion of Li-Yorke chaotic pullback attractor for a process is introduced, and we prove that this chaotic behaviour might occur for almost all the processes. When the problems are additionally sublinear, more cases of forwards attraction are found, which had not been previously considered even in the case of linear-dissipative ODEs.
2020-01-01T00:00:00ZAlmost symmetric numerical semigroups with high typeGarcía Sánchez, Pedro A.Ojeda, Ignaciohttps://uvadoc.uva.es/handle/10324/408102021-05-21T22:23:38Z2019-01-01T00:00:00ZWe establish a one-to-one correspondence between numerical semigroups of genus g and almost symmetric numerical semigroups with Frobenius number F and type F−2g, provided that F is greater than or equal to 4g−1.
2019-01-01T00:00:00ZSemigroups with fixed multiplicity and embedding dimensionGarcía García, Juan IgnacioMarín Aragón, DanielMoreno Frías, María ÁngelesRosales, José CarlosVigneron Tenorio, Albertohttps://uvadoc.uva.es/handle/10324/408092021-05-21T22:23:38Z2019-01-01T00:00:00ZGiven m ∈ N, a numerical semigroup with multiplicity m is called a packed numerical
semigroup if its minimal generating set is included in {m, m + 1, . . . , 2m − 1}. In this
work, packed numerical semigroups are used to build the set of numerical semigroups
with a given multiplicity and embedding dimension, and to create a partition of this set.
Wilf’s conjecture is verified in the tree associated to some packed numerical semigroups.
Furthermore, given two positive integers m and e, some algorithms for computing the
minimal Frobenius number and minimal genus of the set of numerical semigroups with multiplicity m and embedding dimension e are provided. We also compute the semigroups
where these minimal values are achieved
2019-01-01T00:00:00ZBinomial Ideals and Congruences on NnMatusevich, Laura FeliciaOjeda, Ignaciohttps://uvadoc.uva.es/handle/10324/408082021-05-21T22:23:42Z2018-01-01T00:00:00ZA congruence on Nn is an equivalence relation on Nn that is compatible with the additive structure. If k is a field, and I is a binomial ideal in k[X1,…,Xn] (that is, an ideal generated by polynomials with at most two terms), then I induces a congruence on Nn by declaring u and v to be equivalent if there is a linear combination with nonzero coefficients of Xu and Xv that belongs to I. While every congruence on Nn arises this way, this is not a one-to-one correspondence, as many binomial ideals may induce the same congruence. Nevertheless, the link between a binomial ideal and its corresponding congruence is strong, and one may think of congruences as the underlying combinatorial structures of binomial ideals. In the current literature, the theories of binomial ideals and congruences on Nn are developed separately. The aim of this survey paper is to provide a detailed parallel exposition, that provides algebraic intuition for the combinatorial analysis of congruences. For the elaboration of this survey paper, we followed mainly (Kahle and Miller Algebra Number Theory 8(6):1297–1364, 2014) with an eye on Eisenbud and Sturmfels (Duke Math J 84(1):1–45, 1996) and Ojeda and Piedra Sánchez (J Symbolic Comput 30(4):383–400, 2000).
2018-01-01T00:00:00ZOn graph combinatorics to improve eigenvector-based measures of centrality in directed networksArratia, ArgimiroMarijuán López, Carloshttps://uvadoc.uva.es/handle/10324/408072021-05-21T22:23:37Z2016-01-01T00:00:00ZWe present a combinatorial study on the rearrangement of links in the structure of directed networks for the purpose of improving the valuation of a vertex or group of vertices as established by an eigenvector-based centrality measure. We build our topological classification starting from unidirectional rooted trees and up to more complex hierarchical structures such as acyclic digraphs, bidirectional and cyclical rooted trees (obtained by closing cycles on unidirectional trees). We analyze different modifications on the structure of these networks and study their effect on the valuation given by the eigenvector-based scoring functions, with particular focus on α-centrality and PageRank.
2016-01-01T00:00:00ZAre algebraic links in the Poincaré sphere determined by their Alexander polynomials?Campillo López, AntonioDelgado de la Mata, FélixGusein-Zade, Sabir M.https://uvadoc.uva.es/handle/10324/407362021-05-21T22:23:37Z2020-01-01T00:00:00ZThe Alexander polynomial in several variables is defined for links in three-dimensional homology spheres, in particular, in the Poincaré sphere: the intersection of the surface S={(z1,z2,z3)∈C3:z51+z32+z23=0} with the 5-dimensional sphere S5ε={(z1,z2,z3)∈C3:|z1|2+|z2|2+|z3|2=ε2}. An algebraic link in the Poincaré sphere is the intersection of a germ of a complex analytic curve in (S, 0) with the sphere S5ε of radius ε small enough. Here we discuss to which extent the Alexander polynomial in several variables of an algebraic link in the Poincaré sphere determines the topology of the link. We show that, if the strict transform of a curve in (S, 0) does not intersect the component of the exceptional divisor corresponding to the end of the longest tail in the corresponding E8-diagram, then its Alexander polynomial determines the combinatorial type of the minimal resolution of the curve and therefore the topology of the corresponding link. The Alexander polynomial of an algebraic link in the Poincaré sphere is determined by the Poincaré series of the filtration defined by the corresponding curve valuations. (They coincide with each other for a reducible curve singularity and differ by the factor (1−t) for an irreducible one.) We show that, under conditions similar to those for curves, the Poincaré series of a collection of divisorial valuations determines the combinatorial type of the minimal resolution of the collection.
2020-01-01T00:00:00ZFulcrum: Flexible Network Coding for Heterogeneous DevicesLucani, Daniel E.Pedersen, Morten VidebækRuano Benito, DiegoSørensen, Chres W.Fitzek, Frank H. P.Heide, JanusGeil, OlavNguyen, VuReisslein, Martinhttps://uvadoc.uva.es/handle/10324/407352021-05-21T22:23:35Z2018-01-01T00:00:00ZWe introduce Fulcrum, a network coding framework that achieves three seemingly conflicting objectives: 1) to reduce the coding coefficient overhead down to nearly n bits per packet in a generation of n packets; 2) to conduct the network coding using only Galois field GF(2) operations at intermediate nodes if necessary, dramatically reducing computing complexity in the network; and 3) to deliver an end-to-end performance that is close to that of a high-field network coding system for high-end receivers, while simultaneously catering to low-end receivers that decode in GF(2). As a consequence of 1) and 3), Fulcrum has a unique trait missing so far in the network coding literature: providing the network with the flexibility to distribute computational complexity over different devices depending on their current load, network conditions, or energy constraints. At the core of our framework lies the idea of precoding at the sources using an expansion field GF(2 h ), h > 1, to increase the number of dimensions seen by the network. Fulcrum can use any high-field linear code for precoding, e.g., Reed-Solomon or Random Linear Network Coding (RLNC). Our analysis shows that the number of additional dimensions created during precoding controls the trade-off between delay, overhead, and computing complexity. Our implementation and measurements show that Fulcrum achieves similar decoding probabilities as high field RLNC but with encoders and decoders that are an order of magnitude faster.
2018-01-01T00:00:00ZStructural and spectral properties of minimal strong digraphsMarijuán López, CarlosGarcía López, JesúsPozo Coronado, Luis Miguelhttps://uvadoc.uva.es/handle/10324/407342021-05-21T22:23:35Z2016-01-01T00:00:00ZIn this article, we focus on structural and spectral properties of minimal strong
digraphs (MSDs). We carry out a comparative study of properties of MSDs versus
trees. This analysis includes two new properties. The first one gives bounds on
the coefficients of characteristic polynomials of trees (double directed trees), and
conjectures the generalization of these bounds to MSDs. As a particular case, we
prove that the independent coemcient of the characteristic polynomial of a tree or
an MSD must be — 1, 0 or 1. For trees, this fact means that a tree has at most one
perfect matching; for MSDs, it means that an MSD has at most one covering by
disjoint cycles. The property states that every MSD can be decomposed in a rooted
spanning tree and a forest of reversed rooted trees, as factors. In our opinión, the
analogies described suppose a significative change in the traditional point of view
about this class of digraphs.
2016-01-01T00:00:00ZOn pseudo-Frobenius elements of submonoids of NdGarcía García, Juan IgnacioOjeda, IgnacioRosales, José CarlosVigneron Tenorio, Albertohttps://uvadoc.uva.es/handle/10324/407302021-05-21T22:23:34Z2020-01-01T00:00:00ZIn this paper we study those submonoids of Nd with a nontrivial pseudo-Frobenius set. In the affine case, we prove that they are the affine semigroups whose associated algebra over a field has maximal projective dimension possible. We prove that these semigroups are a natural generalization of numerical semigroups and, consequently, most of their invariants can be generalized. In the last section we introduce a new family of submonoids of Nd and using its pseudo-Frobenius elements we prove that the elements in the family are direct limits of affine semigroups.
2020-01-01T00:00:00ZAlmost symmetric numerical semigroups with given Frobenius number and typeBranco, Manuel B.Ojeda, IgnacioRosales, José Carloshttps://uvadoc.uva.es/handle/10324/407282021-05-21T22:23:34Z2019-01-01T00:00:00ZWe give two algorithmic procedures to compute the whole set of almost symmetric numerical semigroups with fixed Frobenius number and type, and the whole set of almost symmetric numerical semigroups with fixed Frobenius number. Our algorithms allow to compute the whole set of almost symmetric numerical semigroups with fixed Frobenius number with similar or even higher efficiency that the known ones. They have been implemented in the GAP [The GAP Group, GAP — Groups, Algorithms and Programming, Version 4.8.6; 2016, https://www.gap-system.org] package NumericalSgps [M. Delgado and P. A. García-Sánchez and J. Morais, “numericalsgps”: A GAP package on numerical semigroups, https://github.com/gap-packages/numericalsgps].
2019-01-01T00:00:00ZUniqueness of limit cycles for quadratic vector fieldsBravo, José LuisFernández, ManuelOjeda, IgnacioSánchez, Fernándohttps://uvadoc.uva.es/handle/10324/407272021-05-21T22:23:33Z2019-01-01T00:00:00ZThis article deals with the study of the number of limit
cycles surrounding a critical point of a quadratic planar vector field,
which, in normal form, can be written as x
′ = a1x − y − a3x
2 + (2a2 +
a5)xy+a6y
2
, y
′ = x+a1y+a2x
2+(2a3+a4)xy−a2y
2
. In particular, we
study the semi-varieties defined in terms of the parameters a1, a2, . . . , a6
where some classical criteria for the associated Abel equation apply.
The proofs will combine classical ideas with tools from computational
algebraic geometry.
2019-01-01T00:00:00ZThe short resolution of a semigroup algebraOjeda, IgnacioVigneron Tenorio, Albertohttps://uvadoc.uva.es/handle/10324/407262021-05-21T22:23:33Z2017-01-01T00:00:00ZThis work generalises the short resolution given by Pisón Casares [‘The short resolution of a lattice ideal’, Proc. Amer. Math. Soc. 131(4) (2003), 1081–1091] to any affine semigroup. We give a characterisation of Apéry sets which provides a simple way to compute Apéry sets of affine semigroups and Frobenius numbers of numerical semigroups. We also exhibit a new characterisation of the Cohen–Macaulay property for simplicial affine semigroups.
2017-01-01T00:00:00ZOn the ideal associated to a linear codeMárquez Corbella, IreneMartínez Moro, EdgarSuárez Canedo, Emiliohttps://uvadoc.uva.es/handle/10324/407252021-05-21T22:23:32Z2016-01-01T00:00:00ZThis article aims to explore the bridge between the algebraic structure of a linear code and the complete decoding process. To this end, we associate a specific binomial ideal I+(C) to an arbitrary linear code. The binomials
involved in the reduced Gr¨obner basis of such an ideal relative to a degreecompatible ordering induce a uniquely defined test-set for the code, and this
allows the description of a Hamming metric decoding procedure. Moreover,
the binomials involved in the Graver basis of I+(C) provide a universal test-set
which turns out to be a set containing the set of codewords of minimal support
of the code.
2016-01-01T00:00:00Z