  Let $X$ be a smooth projective quadric defined over a field of characteristic
2. We prove that in the Chow group of codimension 2 or 3 of $X$ the torsion
subgroup has at most two elements. In codimension 2, we determine precisely
when this torsion subgroup is nontrivial. In codimension 3, we show that there
is no torsion if {$\dim X\ge 11$.} This extends the analogous results in
characteristic different from 2, obtained by Karpenko in the nineteen-nineties.
