New quantum codes from metacirculant graphs via self-dual additive $\mathbb{F}_4$-codes

<p style='text-indent:20px;'>We use symplectic self-dual additive codes over <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{F}_4 $\end{document}</tex-math></inline-formula> obtained from metacirculant graphs to construct, for the first time, <inline-formula><tex-math id="M2">\begin{document}$ \left[\kern-0.15em\left[ {\ell, 0, d} \right]\kern-0.15em\right] $\end{document}</tex-math></inline-formula> qubit codes with parameters <inline-formula><tex-math id="M3">\begin{document}$ (\ell,d) \in \{(78, 20), (90, 21), (91, 22), (93,21),(96,22)\} $\end{document}</tex-math></inline-formula>. Secondary constructions applied to the qubit codes result in many new qubit codes that perform better than the previous best-known.</p>

On one-weight and ACD codes in Zr2 x Zs4 x Zt8

In this paper, one-weight and additive complementary dual (ACD) codes in Zr2 x Zs4 x Zt 8 are studied. Firstly, it is shown that the image of an equidistant Z2Z4Z8-additive code is a binary equidistant code. Then, some properties of the structure and possible weights for one-weight Z2Z4Z8-additive codes are described. Finally, it is given the sufficient conditions for a Z2Z4Z8-additive code to be ACD.

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>

On one-lee weight and two-lee weight $ \mathbb{Z}_2\mathbb{Z}_4[u] $ additive codes and their constructions

<p style='text-indent:20px;'>This paper mainly study <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{Z}_{2}\mathbb{Z}_{4}[u] $\end{document}</tex-math></inline-formula>-additive codes. A Gray map from <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{Z}_{2}^{\alpha}\times\mathbb{Z}_{4}^{\beta}[u] $\end{document}</tex-math></inline-formula> to <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{Z}_{4}^{\alpha+2\beta} $\end{document}</tex-math></inline-formula> is defined, and we prove that is a weight preserving and distance preserving map. A MacWilliams-type identity between the Lee weight enumerator of a <inline-formula><tex-math id="M5">\begin{document}$ \mathbb{Z}_{2}\mathbb{Z}_{4}[u] $\end{document}</tex-math></inline-formula>-additive code and its dual is proved. Some properties of one-weight <inline-formula><tex-math id="M6">\begin{document}$ \mathbb{Z}_{2}\mathbb{Z}_{4}[u] $\end{document}</tex-math></inline-formula>-additive codes and two-weight projective <inline-formula><tex-math id="M7">\begin{document}$ \mathbb{Z}_{2}\mathbb{Z}_{4}[u] $\end{document}</tex-math></inline-formula>-additive codes are discussed. As main results, some construction methods for one-weight and two-weight <inline-formula><tex-math id="M8">\begin{document}$ \mathbb{Z}_{2}\mathbb{Z}_{4}[u] $\end{document}</tex-math></inline-formula>-additive codes are studied, meanwhile several examples are presented to illustrate the methods.</p>