On additive MDS codes over small fields

Abstract

<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>

The equivalence of linear codes implies semi-linear equivalence

“…For example, in a classification of codes up to equivalence, as in [1], [8] and [9], one will not find two linear codes in the same equivalence class. Perhaps of more interest, in a classification of linear codes up to semi-linear equivalence, as in [2] and [5], we can now be sure that two semi-linearly inequivalent codes are also inequivalent.…”

“…This more general equivalence we will call semi-linear equivalence, following [3], [6] and [10]. It is also called PΓL-equivalence [2] and is referred to as simply equivalence in [4] and [7].…”

The geometry of non-additive stabiliser codes

“…This latter property implies that the size of the minimum set of dependent points on distinct lines of X t,u is at least 3, since two points are dependent if and only if they are the same point. One can check that T is a subspace so, by Theorem 6, Q(S, T ) is a [ [11,6,3]] 3 stabiliser code. Furthermore, this is an optimal stabliser code for an [ [11, k, 3]] 3 code since there is no quantum MDS code (a code attaining the quantum Singleton bound) with these parameters.…”

“…Recall that the quantum Singleton bound, proved by Rains in [15], states that k n − 2(d − 1), which in this case gives k 7. However, the existence of an [ [11,7,3]] 3 stabiliser code can be ruled out, since there is no additive MDS code of length 11 over F 9 , see [3].…”

“…For example, as a stabiliser code the optimal [ [5, k, 2]] code is attained by the 4-dimensional [[5, 2, 2]] code. However, as discovered in [16], there is a ((5, 6, 2)) which is the direct sum of six [ [5,0,3]] stabiliser codes. A simple description of this code was given using graphs in [17], which also contained a construction of a ((9, 12, 3)) non-additive stabiliser code.…”