  We show that in $\mathbb{R}^d$ there are purely unrectifiable sets of
Hausdorff (and even box counting) dimension $d-1$ which are not tube null,
settling a question of Carbery, Soria and Vargas, and improving a number of
results by the same authors and by Carbery. Our method extends also to "convex
tube null sets", establishing a contrast with a theorem of Alberti,
Cs\"{o}rnyei and Preiss on Lipschitz-null sets. The sets we construct are
random, and the proofs depend on intersection properties of certain random
fractal measures with curves.
