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Shimada and Zhang studied the existence of polarizations on some supersingular $K3$ surfaces by reducing the existence of the polarizations to that of ternary $[12,5]$ codes satisfying certain conditions. In this note, we give a classification of ternary $[12,5]$ codes satisfying the conditions. To do this, ternary $[10,5]$ codes are classified for minimum weights $3$ and $4$.

On the classification of certain ternary codes of length 12