  We introduce the notion of $su(2)$ spin-$s$ Dicke states, which are
higher-spin generalizations of usual (spin-1/2) Dicke states. These multi-qudit
states can be expressed as superpositions of $su(2s+1)$ qudit Dicke states.
They satisfy a recursion formula, which we use to formulate an efficient
quantum circuit for their preparation, whose size scales as $s k n$ for $k\ll 2
s n$, where $n$ is the number of qudits and $k$ is the number of times the
total spin-lowering operator is applied to the highest-weight state. The
algorithm is deterministic and does not require ancillary qudits.
