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On optimal ternary linear codes of dimension 6

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  • We prove that $[g_3(6,d),6,d]_3$ codes for $d=253$-$267$ and $[g_3(6,d)+1,6,d]_3$ codes for $d=302, 303, 307$-$312$ exist and that $[g_3(6,d),6,d]_3$ codes for $d=175, 200, 302, 303, 308, 309$ and a $[g_3(6,133)+1,6,133]_3$ code do not exist, where $g_3(k,d)=\sum_{i=0}^{k-1} \lceil d / 3^i \rceil$. These determine $n_3(6,d)$ for $d=133, 175, 200, 253$-$267, 302, 303, 308$-$312$, where $n_q(k,d)$ is the minimum length $n$ for which an $[n,k,d]_q$ code exists. The updated $n_3(6,d)$ table is also given.
    Mathematics Subject Classification: Primary: 94B27, 94B05; Secondary: 51E20, 05B25.


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