  The optimum distance profiles of linear block codes were studied for
increasing or decreasing message length while keeping the minimum distances as
large as possible, especially for Golay codes and the second-order Reed-Muller
codes, etc. Cyclic codes have more efficient encoding and decoding algorithms.
In this paper, we investigate the optimum distance profiles with respect to the
cyclic subcode chains (ODPCs) of the punctured generalized second-order
Reed-Muller codes $\mathcal{GRM}(2,m)^*$ which were applied in Power Control in
OFDM Modulations in channels with synchronization, and so on. For this, two
standards are considered in the inverse dictionary order, i.e., for increasing
message length. Four lower bounds and upper bounds on ODPC are presented, where
the lower bounds almost achieve the corresponding upper bounds in some sense.
The discussions are over nonbinary prime field.
