The non-classical 10-arc of PG(4, 9)

AbstractIt is shown that PG(4,9) contains a non-classical 10-arc. It is the first example of a (q + 1)-arc of PG(n, q), (q ood, 2⩽n⩽q −2), which is not a normal rational curve. Various properties of the arc are also derived

Twisted Reed-Solomon Codes

We present a new general construction of MDS codes over a finite field $\mathbb{F}_q$. We describe two explicit subclasses which contain new MDS codes of length at least $q/2$ for all values of $q \ge 11$. Moreover, we show that most of the new codes are not equivalent to a Reed-Solomon code.Comment: 5 pages, accepted at IEEE International Symposium on Information Theory 201

Counting generalized Reed-Solomon codes

In this article we count the number of generalized Reed-Solomon (GRS) codes of dimension k and length n, including the codes coming from a non-degenerate conic plus nucleus. We compare our results with known formulae for the number of 3-dimensional MDS codes of length n=6,7,8,9

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

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

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

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