  This paper addresses the decomposition number problem for spin
representations of symmetric groups in odd characteristic. Our main aim is to
find a combinatorial formula for decomposition numbers in blocks of defect $2$,
analogous to Richards's formula for defect $2$ blocks of symmetric groups.
  By developing a suitable analogue of the combinatorics used by Richards, we
find a formula for the corresponding "$q$-decomposition numbers", i.e.\ the
canonical basis coefficients in the level-$1$ $q$-deformed Fock space of type
$A^{(2)}_{2n}$; a special case of a conjecture of Leclerc and Thibon asserts
that these coefficients yield the spin decomposition numbers in characteristic
$2n+1$. Along the way, we prove some general results on $q$-decomposition
numbers. This paper represents the first substantial progress on canonical
bases in type $A^{(2)}_{2n}$.
