Bounds on the minimum distance of locally recoverable codes

We consider locally recoverable codes (LRCs) and aim to determine the smallest possible length $n=n_q(k,d,r)$ of a linear $[n,k,d]_q$-code with locality $r$. For $k\le 7$ we exactly determine all values of $n_2(k,d,2)$ and for $k\le 6$ we exactly determine all values of $n_2(k,d,1)$. For the ternary field we also state a few numerical results. As a general result we prove that $n_q(k,d,r)$ equals the Griesmer bound if the minimum Hamming distance $d$ is sufficiently large and all other parameters are fixed.Comment: 23 pages, 3 table

Minimal triangulations of sphere bundles over the circle

For integers $d \geq 2$ and $\epsilon = 0$ or 1, let $S^{1, d - 1}(\epsilon)$ denote the sphere product $S^{1} \times S^{d - 1}$ if $\epsilon = 0$ and the twisted $S^{d - 1}$ bundle over $S^{1}$ if $\epsilon = 1$. The main results of this paper are: (a) if $d \equiv \epsilon$ (mod 2) then $S^{1, d - 1}(\epsilon)$ has a unique minimal triangulation using $2d + 3$ vertices, and (b) if $d \equiv 1 - \epsilon$ (mod 2) then $S^{1, d - 1}(\epsilon)$ has minimal triangulations (not unique) using $2d + 4$ vertices. The second result confirms a recent conjecture of Lutz. The first result provides the first known infinite family of closed manifolds (other than spheres) for which the minimal triangulation is unique. Actually, we show that while $S^{1, d - 1}(\epsilon)$ has at most one $(2d + 3)$-vertex triangulation (one if $d \equiv \epsilon$ (mod 2), zero otherwise), in sharp contrast, the number of non-isomorphic $(2d + 4)$-vertex triangulations of these $d$-manifolds grows exponentially with $d$ for either choice of $\epsilon$. The result in (a), as well as the minimality part in (b), is a consequence of the following result: (c) for $d \geq 3$, there is a unique $(2d + 3)$-vertex simplicial complex which triangulates a non-simply connected closed manifold of dimension $d$. This amazing simplicial complex was first constructed by K\"{u}hnel in 1986. Generalizing a 1987 result of Brehm and K\"{u}hnel, we prove that (d) any triangulation of a non-simply connected closed $d$-manifold requires at least $2d + 3$ vertices. The result (c) completely describes the case of equality in (d). The proofs rest on the Lower Bound Theorem for normal pseudomanifolds and on a combinatorial version of Alexander duality.Comment: 15 pages, Revised, To appear in `Journal of Combinatorial Theory, Ser. A

Abundance of 3-planes on real projective hypersurfaces

We show that a generic real projective $n$-dimensional hypersurface of odd degree $d$, such that $4(n-2)=\binom{d+3}3$, contains "many" real 3-planes, namely, in the logarithmic scale their number has the same rate of growth, $d^3\log d$, as the number of complex 3-planes. This estimate is based on the interpretation of a suitable signed count of the 3-planes as the Euler number of an appropriate bundle.Comment: 25 pages, minor typos corrected after proofreadin

The boundary volume of a lattice polytope

For a d-dimensional convex lattice polytope P, a formula for the boundary volume is derived in terms of the number of boundary lattice points on the first \floor{d/2} dilations of P. As an application we give a necessary and sufficient condition for a polytope to be reflexive, and derive formulae for the f-vector of a smooth polytope in dimensions 3, 4, and 5. We also give applications to reflexive order polytopes, and to the Birkhoff polytope.Comment: 21 pages; subsumes arXiv:1002.1908 [math.CO]; to appear in the Bulletin of the Australian Mathematical Societ

Classification of empty 4-simplices and other lattice polytopes

RESUMEN: Un d-politopo es la envolvente convexa de un conjunto finito de puntos en R^d. En particular, si un d-politopo está generado por exactamente d + 1 puntos se dice que es un símplice o un d-símplice. Además, si tomamos los puntos con coordenadas enteras, se dice que el politopo es reticular. A lo largo de esta tesis doctoral se estudian los politopos reticulares y, más concretamente, se estudian dos tipos de estos que son los politopos reticulares vacíos (cuyos únicos puntos reticulares son los vértices) y los politopos reticulares huecos, politopos reticulares que no poseen puntos reticulares en su interior relativo, es decir, todos sus puntos reticulares se encuentran en la frontera. Los politopos huecos, también vacíos, aparecen como el ejemplo más sencillo de politopos reticulares al no tener puntos enteros en el interior de su envolvente convexa. El principal resultado de la tesis doctoral es la clasificación de símplices vacíos en dimensión 4. Mientras los casos en dimensión 1 y 2 son triviales y el caso de dimensión 3 estaba concluido desde 1964 con el trabajo de White [Whi64], con este trabajo se completa esta clasificación en dimensión 4. Artículos como el de Mori, Morrison y Morrison [MMM88] en 1988 consiguen describir algunas familias de 4-símplices vacíos de volumen primo en términos de quíntuplas. Otros trabajos como el de Haase y Ziegler [HZ00] en el 2000, obtienen resultados parciales de esta clasificación. En particular, en ese trabajo se conjeturó una lista completa de 4-símplices vacíos con anchura mayor que dos, la cual se prueba completa en esta tesis. Empleando técnicas de geometría convexa, geometría de números y resultados previos sobre la relación entre la anchura de un politopo y su volumen, somos capaces de establecer unas cotas superiores para los 4-símplices vacíos que deseamos clasificar. Con estas cotas para el volumen de los símplices y una gran cantidad de computación de estos politopos reticulares en dimensión 4 somos capaces de completar la clasificación, explicando el método general utilizado para describir las familias de símplices vacíos que aparecen en la clasificación.ABSTRACT: A d-polytope is the convex hull of a finite set of points in R^d. In particular, if a d-polytope is generated by exactly d + 1 points, it is said to be a simplex or a d-simplex. In addition, if we take the points with integer coordinates, the polytope is a lattice polytope. Throughout this thesis, lattice polytopes are studied and, more specifically, two types of these, which are empty lattice polytopes (whose only integer points are its vertices) and hollow polytopes, lattice polytopes that do not have integer points in their interior, that is, all their integer points are in their facets. Hollow polytopes, also empty, appear as the simplest example of lattice polytopes because they have no integer points inside their convex hull. The main result of the thesis is the classification of empty simplices in dimension 4. While cases in dimension 1 and 2 are trivial and the case of dimension 3 has been completed since 1964 with the work of White [Whi64], this work completes this classification in dimension 4. Papers such as Mori, Morrison and Morrison [MMM88] in 1988 manage to describe some families of empty 4-simplices of prime volume in terms of quintuples. Other works, such as Haase and Ziegler [HZ00] in 2000, obtain partial results tor this classification. In particular, this work conjecture a complete list of empty 4-simplices of width greater than two, which is verified in this thesis. With convex geometry tools, geometry of numbers and previous results that rely on the relationship between the width of a polytope and its volume, we are able to to set upper bounds for the volume of hollow 4-simpolices, that we want to classify. With these upper bounds for the volume of the simplices and a lot of computation of these lattice polytopes in dimension 4 we are able to complete the classification, explaining the general method used to describe the families of empty simplices that appear in the classification.This thesis has been developed under the following scholarships and project grants: MTM2014-54207-P, MTM2017-83750-P and BES-2015-073128 of the Spanish Ministry of Economy and Competitiveness

Complete enumeration of two-Level orthogonal arrays of strength dd with d+2d+2 constraints

Enumerating nonisomorphic orthogonal arrays is an important, yet very difficult, problem. Although orthogonal arrays with a specified set of parameters have been enumerated in a number of cases, general results are extremely rare. In this paper, we provide a complete solution to enumerating nonisomorphic two-level orthogonal arrays of strength $d$ with $d+2$ constraints for any $d$ and any run size $n=\lambda2^d$. Our results not only give the number of nonisomorphic orthogonal arrays for given $d$ and $n$, but also provide a systematic way of explicitly constructing these arrays. Our approach to the problem is to make use of the recently developed theory of $J$-characteristics for fractional factorial designs. Besides the general theoretical results, the paper presents some results from applications of the theory to orthogonal arrays of strength two, three and four.Comment: Published at http://dx.doi.org/10.1214/009053606000001325 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org