On the classification of certain ternary codes of length 12

TL;DR

This paper classifies specific ternary codes of length 12, which are linked to the existence of polarizations on supersingular K3 surfaces, by analyzing related codes of shorter length.

Contribution

It provides a complete classification of ternary [12,5] codes satisfying certain conditions, extending the classification to [10,5] codes for particular minimum weights.

Findings

Classified all ternary [12,5] codes meeting the specified conditions.

Extended classification to ternary [10,5] codes for minimum weights 3 and 4.

Facilitated understanding of polarizations on supersingular K3 surfaces.

Abstract

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 $4$.