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In the present work, we classify sets of type (4,n) in PG(3,q). We prove that PG(3,q), apart from the planes of PG(3,3), contains only sets of type (4,n) with standard parameters. Thus, somewhat surprisingly, we conclude that there are no sets of type (4,n) in PG(3,q), q ≠ 3, with non-standard parameters.
Stefano Innamorati (2024) studied this question.