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

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 sets defining few ordinary solids

The version of record is available online at: http://dx.doi.org/10.1007/s00454-021-00302-7Let 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) then, for n sufficiently large, all but at most O(K) points of 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 that span less than n3/6 solids containing exactly four points of S.Peer ReviewedPostprint (author's final draft

On sets of points with few ordinary hyperplanes

Let $S$ be a set of $n$ points in the projective $d$-dimensional real space $\mathbb{RP}^d$ such that not all points of $S$ are contained in a single hyperplane and such that any subset of $d$ points of $S$ span a hyperplane. Let an ordinary hyperplane of $S$ be an hyperplane of $\mathbb{RP}^d$ containing exactly $d$ points of $S$. In this paper we study the minimum number of ordinary hyperplanes spanned by any set $S$ of $n$ points in $4$ dimensions, following the work of Ben Green and Terence Tao in the planar version of the problem, as well as the work of Simeon Ball in the $3$ dimensional case. We classify the sets of points in $4$ dimensions that span few ordinary hyperplanes, showing that if $S$ is a set spanning less than $Kn^3$ ordinary hyperplanes, for some $K = o(n^{\frac{1}{6}})$, then all but $O(K)$ points of $S$ must be contained in the intersection of $5$ linearly independent quadrics