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Simeon Ball

Publications and source records attributed to Simeon Ball.

At least 37 records · Page 2Linked to original sources
arXiv2021

Determining when a truncated generalised Reed-Solomon code is Hermitian self-orthogonal

We prove that there is a Hermitian self-orthogonal $k$-dimensional truncated generalised Reed-Solomon code of length $n \leqslant q^2$ over ${\mathbb F}_{q^2}$ if and only if there is a polynomial $g \in {\mathbb F}_{q^2}$ of degree at most $(q-k)q-1$ such that $g+g^q$ has $q^2-n$ distinct zeros. This allows us to determine the smallest $n$ for which there is a Hermitian self-orthogonal $k$-dimensional truncated generalised Reed-Solomon code of length $n$ over ${\mathbb F}_{q^2}$, verifying a conjecture of Grassl and R\"otteler. We also provide examples of Hermitian self-orthogonal $k$-dimensional generalised Reed-Solomon codes of length $q^2+1$ over ${\mathbb F}_{q^2}$, for $k=q-1$ and $q$ an odd power of two.

cs.IT↗
arXiv2020

On additive MDS codes over small fields

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 $n \geqslant q+k$ and $k+1$ of the projections of $C$ are linear over ${\mathbb F}_{q^2}$ then $C$ is linear over ${\mathbb F}_{q^2}$. We use this geometrical theorem, other geometric arguments and some computations to classify all additive MDS codes over ${\mathbb F}_q$ for $q \in \{4,8,9\}$. We also classify the longest additive MDS codes over ${\mathbb F}_{16}$ which are linear over ${\mathbb F}_4$. In these cases, the classifications not only verify the MDS conjecture for additive codes, but also confirm there are no additive non-linear MDS codes which perform as well as their linear counterparts. These results imply that the quantum MDS conjecture holds for $q \in \{ 2,3\}$.

cs.IT↗
arXiv2020

Quantum error-correcting codes and their geometries

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 environment. Quantum error-correcting codes allow the negation of these effects in order to successfully restore the original quantum information. We briefly describe the necessary quantum mechanical background to be able to understand how quantum error-correction works. We go on to construct quantum codes: firstly qubit stabilizer codes, then qubit non-stabilizer codes, and finally codes with a higher local dimension. We will delve into the geometry of these codes. This allows one to deduce the parameters of the code efficiently, deduce the inequivalence between codes that have the same parameters, and presents a useful tool in deducing the feasibility of certain parameters. We also include sections on quantum maximum distance separable codes and the quantum MacWilliams identities.

quant-ph↗
arXiv2019

Arcs in finite projective spaces

This is an expository article detailing results concerning large arcs in finite projective spaces, which attempts to cover the most relevant results on arcs, simplifying and unifying proofs of known old and more recent theorems. The article is mostly self-contained and includes a proof of the most general form of Segre's lemma of tangents and a short proof of the MDS conjecture over prime fields based on this lemma.

math.CO↗
arXiv2019

Some constructions of quantum MDS codes

We construct quantum MDS codes with parameters $ [\![ q^2+1,q^2+3-2d,d ]\!] _q$ for all $d \leqslant q+1$, $d \neq q$. 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 $d\geqslant q+2$ then there is no generalised Reed-Solomon $[n,n-d+1,d]_{q^2}$ 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 $d \geqslant q+3$, apart from the $ [\![ 10,0,6 ]\!] _3$ quantum MDS code derived from Glynn's code.

quant-ph↗
arXiv2019

On varieties defined by large sets of quadrics and their application to error-correcting codes

Let $U$ be a $({ k-1 \choose 2}-1)$-dimensional subspace of quadratic forms defined on $\mathrm{PG}(k-1,{\mathbb F})$ with the property that $U$ does not contain any reducible quadratic form. Let $V(U)$ be the points of $\mathrm{PG}(k-1,{\mathbb F})$ which are zeros of all quadratic forms in $U$. We will prove that if there is a group $G$ which fixes $U$ and no line of $\mathrm{PG}(k-1,{\mathbb F})$ and $V(U)$ spans $\mathrm{PG}(k-1,{\mathbb F})$ then any hyperplane of $\mathrm{PG}(k-1,{\mathbb F})$ is incident with at most $k$ points of $V(U)$. If ${\mathbb F}$ is a finite field then the linear code generated by the matrix whose columns are the points of $V(U)$ is a $k$-dimensional linear code of length $|V(U)|$ and minimum distance at least $|V(U)|-k$. A linear code with these parameters is an MDS code or an almost MDS code. We will construct examples of such subspaces $U$ and groups $G$, 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 $V(U)$ from any $k-4$ 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.

math.CO↗
arXiv2019

Arcs and tensors

To an arc $\mathcal{A}$ of $\mathrm{PG}(k-1,q)$ of size $q+k-1-t$ we associate a tensor in $\langle \nu_{k,t}(\mathcal{A})\rangle^{\otimes k-1}$, where $\nu_{k,t}$ denotes the Veronese map of degree $t$ defined on $\mathrm{PG}(k-1,q)$. As a corollary we prove that for each arc $\mathcal{A}$ in $\mathrm{PG}(k-1,q)$ of size $q+k-1-t$, which is not contained in a hypersurface of degree $t$, there exists a polynomial $F(Y_1,\ldots,Y_{k-1})$ (in $k(k-1)$ variables) where $Y_j=(X_{j1},\ldots,X_{jk})$, which is homogeneous of degree $t$ in each of the $k$-tuples of variables $Y_j$, which upon evaluation at any $(k-2)$-subset $S$ of the arc $\mathcal{A}$ gives a form of degree $t$ on $\mathrm{PG}(k-1,q)$ whose zero locus is the tangent hypersurface of $\mathcal{A}$ at $S$, i.e. the union of the tangent hyperplanes of $\mathcal{A}$ at $S$. This generalises the equivalent result for planar arcs ($k=3$), proven in \cite{BaLa2018}, to arcs in projective spaces of arbitrary dimension. A slightly weaker result is obtained for arcs in $\mathrm{PG}(k-1,q)$ of size $q+k-1-t$ which are contained in a hypersurface of degree $t$. We also include a new proof of the Segre-Blokhuis-Bruen-Thas hypersurface associated to an arc of hyperplanes in $\mathrm{PG}(k-1,q)$.

math.CO↗
arXiv2018

On sets defining few ordinary solids

Let $\mathcal{S}$ 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 $\mathcal{S}$ is less than $Kn^3$ for some $K=o(n^{\frac{1}{7}})$ then, for $n$ sufficiently large, all but at most $O(K)$ points of $\mathcal{S}$ are contained in the intersection of five linearly independent quadrics. Conversely, we prove that there are finite subgroups of size $n$ of an elliptic curve which span less than $\frac{1}{6}n^3$ solids containing exactly four points of $\mathcal{S}$.

math.MG↗
arXiv2017

On sets of points with few odd secants

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 of the set.

math.CO↗
arXiv2017

Planar arcs

Let $p$ denote the characteristic of ${\mathbb F}_q$, the finite field with $q$ elements. We prove that if $q$ is odd then an arc of size $q+2-t$ in the projective plane over ${\mathbb F}_q$, which is not contained in a conic, is contained in the intersection of two curves, which do not share a common component, and have degree at most $t+p^{\lfloor \log_p t \rfloor}$, provided a certain technical condition on $t$ is satisfied. This implies that if $q$ is odd then an arc of size at least $q-\sqrt{q}+\sqrt{q}/p+3$ is contained in a conic if $q$ is square and an arc of size at least $q-\sqrt{q}+\frac{7}{2}$ is contained in a conic if $q$ is prime. This is of particular interest in the case that $q$ is an odd square, since then there are examples of arcs, not contained in a conic, of size $q-\sqrt{q}+1$, and it has long been conjectured that if $q \neq 9$ is an odd square then any larger arc is contained in a conic. These bounds improve on previously known bounds when $q$ is an odd square and for primes less than $1783$. The previously known bounds, obtained by Segre \cite{Segre1967}, Hirschfeld and Korchm\'aros \cite{HK1996} \cite{HK1998}, and Voloch \cite{Voloch1990b} \cite{Voloch1991}, rely on results on the number of points on algebraic curves over finite fields, in particular the Hasse-Weil theorem and the St\"ohr-Voloch theorem, and are based on Segre's idea to associate an algebraic curve in the dual plane containing the tangents to an arc. In this paper we do not rely on such theorems, but use a new approach starting from a scaled coordinate-free version of Segre's lemma of tangents. Arcs in the projective plane over ${\mathbb F}_q$ of size $q$ and $q+1$, $q$ odd, were classified by Segre \cite{Segre1955b} in 1955. In this article, we complete the classification of arcs of size $q-1$ and $q-2$.

math.CO↗
arXiv2016

A generalisation of Sylvester's problem to higher dimensions

In this article we consider $S$ to be a set of points in $d$-space with the property that any $d$ points of $S$ span a hyperplane and not all the points of $S$ are contained in a hyperplane. The aim of this article is to introduce the function $e_d(n)$, which denotes the minimal number of hyperplanes meeting $S$ in precisely $d$ points, minimising over all such sets of points $S$ with $|S|=n$.

math.MG↗
arXiv2016

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 $Kn^2$ for some $K=o(n^{\frac{1}{7}})$ 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 $\frac{1}{2}n^2-cn$ then, for $n$ sufficiently large, $S$ is either a prism, an anti-prism, a prism with a point removed or an 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 $Kn^2$ for some $K=o(n^{\frac{1}{7}})$ then, for $n$ sufficiently large, all but at most $O(K)$ points of $S$ are contained in a curve of degree at most four.

math.MG↗
arXiv2016

On subsets of the normal rational curve

A normal rational curve of the $(k-1)$-dimensional projective space over ${\mathbb F}_q$ is an arc of size $q+1$, since any $k$ points of the curve span the whole space. In this article we will prove that if $q$ is odd then a subset of size $3k-6$ of a normal rational curve cannot be extended to an arc of size $q+2$. In fact, we prove something slightly stronger. Suppose that $q$ is odd and $E$ is a $(2k-3)$-subset of an arc $G$ of size $3k-6$. If $G$ projects to a subset of a conic from every $(k-3)$-subset of $E$ then $G$ cannot be extended to an arc of size $q+2$. Stated in terms of error-correcting codes we prove that a $k$-dimensional linear maximum distance separable code of length $3k-6$ over a field ${\mathbb F}_q$ of odd characteristic, which can be extended to a Reed-Solomon code of length $q+1$, cannot be extended to a linear maximum distance separable code of length $q+2$.

math.CO↗
arXiv2016

Extending small arcs to large arcs

An arc is a set of vectors of the $k$-dimensional vector space over the finite field with $q$ elements ${\mathbb F}_q$, 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.

math.CO↗
arXiv2015

A finite version of the Kakeya problem

Let $L$ be a set of lines of an affine space over a field and let $S$ be a set of points with the property that every line of $L$ is incident with at least $N$ points of $S$. Let $D$ be the set of directions of the lines of $L$ considered as points of the projective space at infinity. We give a geometric construction of a set of lines $L$, where $D$ contains an $N^{n-1}$ grid and where $S$ has size $2((1/2)N)^n$, 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 $S$ dependent on the ideal generated by the homogeneous polynomials vanishing on $D$. This bound is maximised as $((1/2)N)^n$ plus smaller order terms, for $n\geqslant 4$, when $D$ contains the points of a $N^{n-1}$ grid.

math.CO↗
arXiv2015

Forbidden subgraphs in the norm graph

We show that the norm graph constructed in [J. Koll\'{a}r, L. R\'{o}nyai and T. Szab\'o, Norm-graphs and bipartite Tur\'{a}n numbers, Combinatorica, 16 (1996) 399--406] with $n$ vertices about $\frac{1}{2}n^{2-1/t}$ edges, which contains no copy of $K_{t,(t-1)!+1}$, does not contain a copy of $K_{t+1,(t-1)!-1}$.

math.CO↗
arXiv2014

On the representability of the bi-uniform matroid

Every bi-uniform 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 every given bi-uniform matroid has not been proved. The interest of these problems is due to their implications to secret sharing. The existence of efficient methods to find representations for all bi-uniform matroids is proved here for the first time. The previously known efficient constructions apply only to a particular class of bi-uniform matroids, while the known general constructions were not proved to be efficient. In addition, our constructions provide in many cases representations over smaller finite fields.

math.CO↗