171 research outputs found
Extending small arcs to large arcs
This is a post-peer-review, pre-copyedit version of an article published in European Journal of Mathematics. The final authenticated version is available online at: https://doi.org/10.1007/s40879-017-0193-xAn 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. Given an arc G in a space of odd characteristic, we prove that there is an upper bound on the largest arc containing G. The bound is not an explicit bound but is obtained by computing properties of a matrix constructed from G. In some cases we can also determine the largest arc containing G, or at least determine the hyperplanes which contain exactly k-2 vectors of the large arc. The theorems contained in this article may provide new tools in the computational classification and construction of large arcs.Postprint (author's final draft
On sets of points with few odd secants
We prove that, for odd, a set of points in the projective plane
over the field with elements has at least odd secants, where is
a constant and an odd secant is a line incident with an odd number of points of
the set.Comment: Revised versio
On varieties defined by large sets of quadrics and their application to error-correcting codes
Let be a -dimensional subspace of quadratic forms
defined on with the property that does not
contain any reducible quadratic form. Let be the points of
which are zeros of all quadratic forms in .
We will prove that if there is a group which fixes and no line of
and spans
then any hyperplane of is incident with at most
points of . If is a finite field then the linear code
generated by the matrix whose columns are the points of is a
-dimensional linear code of length and minimum distance at least
. A linear code with these parameters is an MDS code or an almost MDS
code. We will construct examples of such subspaces and groups , which
include the normal rational curve, the elliptic curve, Glynn's arc from
\cite{Glynn1986} and other examples found by computer search. We conjecture
that the projection of from any points is contained in the
intersection of two quadrics, the common zeros of two linearly independent
quadratic forms. This would be a strengthening of a classical theorem of Fano,
which itself is an extension of a theorem of Castelnuovo, for which we include
a proof using only linear algebra
A finite version of the Kakeya problem
Let be a set of lines of an affine space over a field and let be a
set of points with the property that every line of is incident with at
least points of . Let be the set of directions of the lines of
considered as points of the projective space at infinity. We give a geometric
construction of a set of lines , where contains an grid and
where has size , given a starting configuration in the plane.
We provide examples of such starting configurations for the reals and for
finite fields. Following Dvir's proof of the finite field Kakeya conjecture and
the idea of using multiplicities of Dvir, Kopparty, Saraf and Sudan, we prove a
lower bound on the size of dependent on the ideal generated by the
homogeneous polynomials vanishing on . This bound is maximised as
plus smaller order terms, for , when contains
the points of a grid.Comment: A few minor changes to previous versio
Some constructions of quantum MDS codes
We construct quantum MDS codes with parameters for all , . These codes are shown to exist by
proving that there are classical generalised Reed-Solomon codes which contain
their Hermitian dual. These constructions include many constructions which were
previously known but in some cases these codes are new. We go on to prove that
if then there is no generalised Reed-Solomon
code which contains its Hermitian dual. We also construct
an quantum MDS code, an quantum
MDS code and a quantum MDS code, which are the first
quantum MDS codes discovered for which , apart from the quantum MDS code derived from Glynn's code
On sets defining few ordinary planes
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 (Formula presented.) then, for n sufficiently large, all but at most O(K) points of S are contained in the intersection of two quadrics. Furthermore, we prove that there is a constant c such that if the number of planes incident with exactly three points of S is less than (Formula presented.) then, for n sufficiently large, S is either a regular prism, a regular anti-prism, a regular prism with a point removed or a regular anti-prism with a point removed. As a corollary to the main result, we deduce the following theorem. Let S be a set of n points in the real plane. If the number of circles incident with exactly three points of S is less than (Formula presented.) for some (Formula presented.) then, for n sufficiently large, all but at most O(K) points of S are contained in a curve of degree at most four.Postprint (updated version
Grassl–Rötteler cyclic and consta-cyclic MDS codes are generalised Reed–Solomon codes
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 as an addendum to Grassl and Rötteler International Symposium on Information Theory (ISIT), pp 1104–1108 (2015). It can also be considered as an appendix to Ball and Vilar IEEE Trans Inform Theory 68:3796–3805, (2022) where Conjecture 11 of International Symposium on Information Theory (ISIT), pp 1104–1108 (2015), which was stated for Grassl–Rötteler codes, is proven for generalised Reed–Solomon codes. The content of this note, together with IEEE Trans Inform Theory 68:3796–3805, (2022) therefore implies that Conjecture 11 from International Symposium on Information Theory (ISIT), pp. 1104–1108 (2015) is true.Peer ReviewedPostprint (author's final draft
Some constructions of quantum MDS codes
The version of record os available online at: http://dx.doi.org/10.1007/s10623-021-00846-yWe construct quantum MDS codes with parameters [[q2+1,q2+3-2d,d]]q for all d¿q+1, d¿q. These codes are shown to exist by proving that there are classical generalised Reed–Solomon codes which contain their Hermitian dual. These constructions include many constructions which were previously known but in some cases these codes are new. We go on to prove that if d¿q+2 then there is no generalised Reed–Solomon [n,n-d+1,d]q2 code which contains its Hermitian dual. We also construct an [[18,0,10]]5 quantum MDS code, an [[18,0,10]]7 quantum MDS code and a [[14,0,8]]5 quantum MDS code, which are the first quantum MDS codes discovered for which d¿q+3, apart from the [[10,0,6]]3 quantum MDS code derived from Glynn’s code.Postprint (author's final draft
On Segre's Lemma of Tangents
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 sets, sets with few
odd secants and further results on arcs. Here, we survey some of these results
and report on how re-formulations of Segre's lemma of tangents are leading to new
results
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