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# On additive MDS codes over small fields

• * Corresponding author: Simeon Ball

The first author acknowledges the support of the project MTM2017-82166-P of the Spanish Ministerio de Ciencia y Innovación

• Let $C$ be a $(n,q^{2k},n-k+1)_{q^2}$ additive MDS code which is linear over ${\mathbb F}_q$. We prove that if $n \geq q+k$ and $k+1$ of the projections of $C$ are linear over ${\mathbb F}_{q^2}$ then $C$ is linear over ${\mathbb F}_{q^2}$. We use this geometrical theorem, other geometric arguments and some computations to classify all additive MDS codes over ${\mathbb F}_q$ for $q \in \{4,8,9\}$. We also classify the longest additive MDS codes over ${\mathbb F}_{16}$ which are linear over ${\mathbb F}_4$. 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 $q \in \{ 2,3\}$.

Mathematics Subject Classification: Primary: 94B27; Secondary: 51E22.

 Citation: • • Table 1.  The classification of arcs of lines of $\mathrm{PG}(5, 2)$

 size 4 5 6 number of arcs of points of $\mathrm{PG}(2, 4)$ 1 1 1 number of arcs of lines of $\mathrm{PG}(5, 2)$ 1 1 1

Table 2.  The classification of arcs of planes of PG(8, 2)

 size 4 5 6 7 8 9 10 number of arcs of points of PG(2, 8) 1 1 3 2 2 2 1 number of arcs of planes of PG(8, 2) 1 2 4 2 2 2 1

Table 3.  The classification of arcs of lines of PG(5, 3)

 size 4 5 6 7 8 9 10 # of arcs of points of PG(2, 9) 1 2 6 3 2 1 1 # of arcs of lines of PG(5, 3) 1 4 13 4 3 1 1

Table 4.  The classification of arcs of lines of PG(3, 3)

 size 4 5 6 7 8 9 10 # of arcs of points of PG(1, 9) 2 2 2 1 1 1 1 # of arcs of lines of PG(3, 3) 3 4 5 4 3 2 2

Table 5.  The classification of arcs of lines of $\mathrm{PG}(5, 4)$

 size 5 6 7 8 9 10 11 # of arcs of $\mathrm{PG}(2, 16)$ 3 22 125 865 1534 1262 300 # of line-arcs of $\mathrm{PG}(5, 4)$ 10 360 8294 15162 2869 1465 301 size 12 13 14 15 16 17 18 # of arcs of $\mathrm{PG}(2, 16)$ 159 70 30 9 5 3 2 # of line-arcs of $\mathrm{PG}(5, 4)$ 159 70 30 9 5 3 2
• Tables(5)

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