Existence of the implicated ternary six-dimensional Griesmer codes

Determine whether the hypothetical ternary Griesmer codes of dimension 6 implicated by the characterization of (70, 22)-minihypers in PG(4, 3) exist or do not exist.

Background

The paper characterizes every (70, 22)-minihyper in PG(4, 3) as either the sum of a solid and a (30, 9)-minihyper or the sum of a (66, 21)-minihyper and a line. The authors explain that this structural result is intended to support attacks on the existence of certain ternary linear codes of dimension 6. The existence status of these codes is explicitly described as unresolved, making their existence or nonexistence an open problem connected to the paper’s minihyper classification.

References

This characterization is crucial for attacking the nonexistence of some ternary 6-dimensional codes whose existence is in doubt (cf [7]).

— A new reducibility results for minihypers in finite projective geometries  (2502.13271 - Landjev et al., 18 Feb 2025) in Section 4, page 6