Ara es mostren els items 31-50 de 55

    • Extending small arcs to large arcs 

      Ball, Simeon Michael (2018-03-01)
      Article
      Accés obert
      An arc is a set of vectors of the k-dimensional vector space over the finite field with q elements Fq , in which every subset of size k is a basis of the space, i.e. every k-subset is a set of linearly independent vectors. ...
    • Forbidden subgraphs in the norm graph 

      Ball, Simeon Michael; Pepe, Valentina (2016-04-06)
      Article
      Accés obert
      We show that the norm graph with n vertices about View the MathML source edges, which contains no copy of the complete bipartite graph Kt,(t-1)!+1, does not contain a copy of Kt+1,(t-1)!-1.
    • Grassl–Rötteler cyclic and consta-cyclic MDS codes are generalised Reed–Solomon codes 

      Ball, Simeon Michael (Springer, 2022-12-31)
      Article
      Accés obert
      We prove that the cyclic and constacyclic codes constructed by Grassl and Rötteler in International Symposium on Information Theory (ISIT), pp. 1104–1108 (2015) are generalised Reed–Solomon codes. This note can be considered ...
    • On additive MDS codes over small fields 

      Ball, Simeon Michael; Gamboa Jimenez, Gonzalo; Lavrauw, Michel (American Institute of Mathematical Sciences, 2022)
      Article
      Accés obert
      Let $ C $ be a $ (n,q^{2k},n-k+1)_{q^2} $ additive MDS code which is linear over $ {\mathbb F}_q $. We prove that if $ n \geq q+k $ and $ k+1 $ of the projections of $ C $ are linear over $ {\mathbb F}_{q^2} $ then $ C $ ...
    • On additive MDS codes with linear projections 

      Adriaensen, Sam; Ball, Simeon Michael (Elsevier, 2023-10-01)
      Article
      Accés obert
      We support some evidence that a long additive MDS code over a finite field must be equivalent to a linear code. More precisely, let C be an -linear MDS code over . If , , , and C has three coordinates from which its ...
    • On Segre's lemma of tangents 

      Ball, Simeon Michael; Csajbók, Bence (2018-07-01)
      Article
      Accés obert
      Segre’s lemma of tangents dates back to the 1950’s when he used it in the proof of his “arc is a conic” theorem. Since then it has been used as a tool to prove results about various objects including internal nuclei, Kakeya ...
    • On sets defining few ordinary planes 

      Ball, Simeon Michael (2017-09-19)
      Article
      Accés obert
      Let S be a set of n points in real three-dimensional space, no three collinear and not all co-planar. We prove that if the number of planes incident with exactly three points of S is less than (Formula presented.) for some ...
    • On sets defining few ordinary solids 

      Ball, Simeon Michael; Jiménez Izquierdo, Enrique (Springer Nature, 2021-05-04)
      Article
      Accés obert
      Let S be a set of n points in real four-dimensional space, no four coplanar and spanning the whole space. We prove that if the number of solids incident with exactly four points of S is less than Kn3 for some K=o(n1/7) ...
    • On sets of points with few odd secants 

      Ball, Simeon Michael; Csajbok, Bence (2019-10-10)
      Article
      Accés obert
      We prove that, for q odd, a set of q + 2 points in the projective plane over the field with q elements has at least 2q - c odd secants, where c is a constant and an odd secant is a line incident with an odd number of points ...
    • On sets of vectors of a finite vector space in which every subset of basis size is a basis II 

      Ball, Simeon Michael; De Beule, Jan (2012-10)
      Article
      Accés obert
      This article contains a proof of the MDS conjecture for k = 2p - 2. That is, that if S is a set of vectors of F k q in which every subset of S of size k is a basis, where q = p h, p is prime and q is not and k = 2p - 2, ...
    • On subsets of the normal rational curve 

      Ball, Simeon Michael; De Beule, Jan (2017-06-01)
      Article
      Accés obert
      A normal rational curve of the (k-1) -dimensional projective space over Fq is an arc of size q+1 , since any k points of the curve span the whole space. In this paper, we will prove that if q is odd, then a subset of size ...
    • On the graph of a function over a prime field whose small powers have bounded degree 

      Ball, Simeon Michael; Gács, Andras (2007-11)
      Article
      Accés obert
      Let $f$ be a function from a finite field ${\mathbb F}_p$ with a prime number $p$ of elements, to ${\mathbb F}_p$. In this article we consider those functions $f(X)$ for which there is a positive integer $n > 2\sqrt{p-1 ...
    • On the representability of the biuniform matroid 

      Ball, Simeon Michael; Padró Laimon, Carles; Weiner, Zsuzsa; Xing, Chaoping (2013-08-22)
      Article
      Accés obert
      Every biuniform matroid is representable over all sufficiently large fields. But it is not known exactly over which finite fields they are representable, and the existence of efficient methods to find a representation for ...
    • On varieties defined by large sets of quadrics and their application to error-correcting codes 

      Ball, Simeon Michael; Pepe, Valentina (2020-10-01)
      Article
      Accés obert
      Let U be a ( k-1 2 - 1)-dimensional subspace of quadratic forms defined on F k with the property that U does not contain any reducible quadratic form. Let V (U) be the points of PG(k - 1, F) which are zeros of all quadratic ...
    • Planar arcs 

      Ball, Simeon Michael; Lavrauw, Michel (2018-11-01)
      Article
      Accés obert
      Let p denote the characteristic of , the finite field with q elements. We prove that if q is odd then an arc of size in the projective plane over , which is not contained in a conic, is contained in the intersection of two ...
    • Polynomials in finite geometries 

      Ball, Simeon Michael (Cambridge University Press, 1999)
      Comunicació de congrés
      Accés obert
      A method of using polynomials to describe objects in finite geometries is outlined and the problems where this method has led to a solution are surveyed. These problems concern nuclei, affine blocking sets, maximal arcs ...
    • Polynomials in finite geometry 

      Ball, Simeon Michael; Blokhuis, Aart (Università degli studi di Napoli. Dipartimento di Matematica, 1999-11-01)
      Capítol de llibre
      Accés obert
    • Punctured combinatorial Nullstellensätze 

      Ball, Simeon Michael; Serra Albó, Oriol (2007-11)
      Article
      Accés obert
      In this article we present a punctured version of Alon's Nullstellensatz which states that if $f$ vanishes at nearly all, but not all, of the common zeros of some polynomials $g_1(X_1),\ldots,g_n(X_n)$ then every $I$-residue ...
    • QUANTUM COMPUTATION 

      Ball, Simeon Michael (Universitat Politècnica de Catalunya, 2021-04-01)
      Examen
      Accés restringit a la comunitat UPC
    • Quantum error-correcting codes and their geometries 

      Ball, Simeon Michael; Centelles Tarrés, Aina; Huber, Felix (European Mathematical Society (EMS), 2023-02-10)
      Article
      Accés obert
      This is an expository article aiming to introduce the reader to the underlying mathematics and geometry of quantum error correction. Information stored on quantum particles is subject to noise and interference from the ...