On additive MDS codes over small fields

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ C $\end{document}</tex-math></inline-formula> be a <inline-formula><tex-math id="M2">\begin{document}$ (n,q^{2k},n-k+1)_{q^2} $\end{document}</tex-math></inline-formula> additive MDS code which is linear over <inline-formula><tex-math id="M3">\begin{document}$ {\mathbb F}_q $\end{document}</tex-math></inline-formula>. We prove that if <inline-formula><tex-math id="M4">\begin{document}$ n \geq q+k $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ k+1 $\end{document}</tex-math></inline-formula> of the projections of <inline-formula><tex-math id="M6">\begin{document}$ C $\end{document}</tex-math></inline-formula> are linear over <inline-formula><tex-math id="M7">\begin{document}$ {\mathbb F}_{q^2} $\end{document}</tex-math></inline-formula> then <inline-formula><tex-math id="M8">\begin{document}$ C $\end{document}</tex-math></inline-formula> is linear over <inline-formula><tex-math id="M9">\begin{document}$ {\mathbb F}_{q^2} $\end{document}</tex-math></inline-formula>. We use this geometrical theorem, other geometric arguments and some computations to classify all additive MDS codes over <inline-formula><tex-math id="M10">\begin{document}$ {\mathbb F}_q $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M11">\begin{document}$ q \in \{4,8,9\} $\end{document}</tex-math></inline-formula>. We also classify the longest additive MDS codes over <inline-formula><tex-math id="M12">\begin{document}$ {\mathbb F}_{16} $\end{document}</tex-math></inline-formula> which are linear over <inline-formula><tex-math id="M13">\begin{document}$ {\mathbb F}_4 $\end{document}</tex-math></inline-formula>. In these cases, the classifications not only verify the MDS conjecture for additive codes, but also confirm there are no additive non-linear MDS codes which perform as well as their linear counterparts. These results imply that the quantum MDS conjecture holds for <inline-formula><tex-math id="M14">\begin{document}$ q \in \{ 2,3\} $\end{document}</tex-math></inline-formula>.</p>

Quantum Codes of Maximal Distance and Highly Entangled Subspaces

We present new bounds on the existence of general quantum maximum distance separable codes (QMDS): the length n of all QMDS codes with local dimension D and distance d≥3 is bounded by n≤D2+d−2. We obtain their weight distribution and present additional bounds that arise from Rains' shadow inequalities. Our main result can be seen as a generalization of bounds that are known for the two special cases of stabilizer QMDS codes and absolutely maximally entangled states, and confirms the quantum MDS conjecture in the special case of distance-three codes. As the existence of QMDS codes is linked to that of highly entangled subspaces (in which every vector has uniform r-body marginals) of maximal dimension, our methods directly carry over to address questions in multipartite entanglement.

Inclusion Matrices and the MDS Conjecture

Let $\mathbb{F}_{q}$ be a finite field of order $q$ with characteristic $p$. An arc is an ordered family of at least $k$ vectors in $\mathbb{F}_{q}^{k}$ in which every subfamily of size $k$ is a basis of $\mathbb{F}_{q}^{k}$. The MDS conjecture, which was posed by Segre in 1955, states that if $k \leq q$, then an arc in $\mathbb{F}_{q}^{k}$ has size at most $q+1$, unless $q$ is even and $k=3$ or $k=q-1$, in which case it has size at most $q+2$. We propose a conjecture which would imply that the MDS conjecture is true for almost all values of $k$ when $q$ is odd. We prove our conjecture in two cases and thus give simpler proofs of the MDS conjecture when $k \leq p$, and if $q$ is not prime, for $k \leq 2p-2$. To accomplish this, given an arc $G \subset \mathbb{F}_{q}^{k}$ and a nonnegative integer $n$, we construct a matrix $M_{G}^{\uparrow n}$, which is related to an inclusion matrix, a well-studied object in combinatorics. Our main results relate algebraic properties of the matrix $M_{G}^{\uparrow n}$ to properties of the arc $G$ and may provide new tools in the computational classification of large arcs.

PURE ASYMMETRIC QUANTUM MDS CODES FROM CSS CONSTRUCTION: A COMPLETE CHARACTERIZATION

Using the Calderbank–Shor–Steane (CSS) construction, pure q-ary asymmetric quantum error-correcting codes attaining the quantum Singleton bound are constructed. Such codes are called pure CSS asymmetric quantum maximum distance separable (AQMDS) codes. Assuming the validity of the classical maximum distance separable (MDS) Conjecture, pure CSS AQMDS codes of all possible parameters are accounted for.