  A point set $\mathrm X_N$ on the unit sphere is a spherical $t$-design is
equivalent to the nonnegative quantity $A_{N,t+1}$ vanished. We show that if
$\mathrm X_N$ is a stationary point set of $A_{N,t+1}$ and the minimal singular
value of basis matrix is positive, then $\mathrm X_N$ is a spherical
$t$-design. Moreover, the numerical construction of spherical $t$-designs is
valid by using Barzilai-Borwein method. We obtain numerical spherical
$t$-designs with $t+1$ up to $127$ at $N=(t+2)^2$.
