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 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

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 $|S|=n$.Postprint (published version

Forbidden subgraphs in the norm graph

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.Postprint (author's final draft

A polynomial-time reduction from the multi-graph isomorphism problem to additive codeequivalence

We present a polynomial-time reduction from the multi-graph isomorphism problem to the problem of code equivalence of additive codes over finite extensions of the field with two elements.Peer ReviewedPostprint (published version

Additive MDS codes

We prove that an additive code over a finite field which has a few projections which are equivalent to a linear code is itself equivalent to a linear code, providing the code is not too short.Postprint (published version

Arcs and tensors

This is a post-peer-review, pre-copyedit version of an article published in Designs, Codes, and Cryptography. The final authenticated version is available online at: http://dx.doi.org/10.1007/s10623-019-00668-zTo an arc A of PG(k-1,q) of size q+k-1-t we associate a tensor in ¿¿k,t(A)¿¿k-1 , where ¿k,t denotes the Veronese map of degree t defined on PG(k-1,q) . As a corollary we prove that for each arc A in 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(Y1,…,Yk-1) (in k(k-1) variables) where Yj=(Xj1,…,Xjk) , which is homogeneous of degree t in each of the k-tuples of variables Yj , which upon evaluation at any (k-2) -subset S of the arc A gives a form of degree t on PG(k-1,q) whose zero locus is the tangent hypersurface of A at S, i.e. the union of the tangent hyperplanes of A at S. This generalises the equivalent result for planar arcs ( k=3 ), proven in [2], to arcs in projective spaces of arbitrary dimension. A slightly weaker result is obtained for arcs in 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 PG(k-1,q) .Postprint (author's final draft

Polynomials in finite geometry

Postprint (published version