  We apply the generalized sphere-packing bound to two classes of
subblock-constrained codes. A la Fazeli et al. (2015), we made use of
automorphism to significantly reduce the number of variables in the associated
linear programming problem. In particular, we study binary constant
subblock-composition codes (CSCCs), characterized by the property that the
number of ones in each subblock is constant, and binary subblock
energy-constrained codes (SECCs), characterized by the property that the number
of ones in each subblock exceeds a certain threshold. For CSCCs, we show that
the optimization problem is equivalent to finding the minimum of $N$ variables,
where $N$ is independent of the number of subblocks. We then provide
closed-form solutions for the generalized sphere-packing bounds for single- and
double-error correcting CSCCs. For SECCs, we provide closed-form solutions for
the generalized sphere-packing bounds for single errors in certain special
cases. We also obtain improved bounds on the optimal asymptotic rate for CSCCs
and SECCs, and provide numerical examples to highlight the improvement.
