For nonzero binary b, b dot s is the parity of selected coordinates. Flipping one selected bit pairs opposite signs, so independent uniform shifts give fair independent signs. Require |sum_j (-1)^(b dot s_j)|<=epsilon d, epsilon=1/(200Bmu). Write each sign as 2X_j-1; Hoeffding for d fair zero-one trials with deviation epsilon d/2 gives failure at most 2 exp(-epsilon^2 d/2). If q=1 no condition remains. Otherwise d=D0, and the union bound over fewer than q nonzero b gives 2q exp(-4mu)<1, since q<2m<2^(mu+1). Enumerating tuples and checking integer-scaled inequalities therefore finds one. Store only the current tuple; the final construction is deterministic.