  The study of subblock-constrained codes has recently gained attention due to
their application in diverse fields. We present bounds on the size and
asymptotic rate for two classes of subblock-constrained codes. The first class
is binary constant subblock-composition codes (CSCCs), where each codeword is
partitioned into equal sized subblocks, and every subblock has the same fixed
weight. The second class is binary subblock energy-constrained codes (SECCs),
where the weight of every subblock exceeds a given threshold. We present novel
upper and lower bounds on the code sizes and asymptotic rates for binary CSCCs
and SECCs. For a fixed subblock length and small relative distance, we show
that the asymptotic rate for CSCCs (resp. SECCs) is strictly lower than the
corresponding rate for constant weight codes (CWCs) (resp. heavy weight codes
(HWCs)). Further, for codes with high weight and low relative distance, we show
that the asymptotic rates for CSCCs is strictly lower than that of SECCs, which
contrasts that the asymptotic rate for CWCs is equal to that of HWCs. We also
provide a correction to an earlier result by Chee et al. (2014) on the
asymptotic CSCC rate. Additionally, we present several numerical examples
comparing the rates for CSCCs and SECCs with those for constant weight codes
and heavy weight codes.
