  The distance profiles of linear block codes can be employed to design
variational coding scheme for encoding message with variational length and
getting lower decoding error probability by large minimum Hamming distance. %,
e.g. the design of TFCI in CDMA and the researches on the second-order
Reed-Muller code $\mathcal{RM}(2,m)$, etc.
  Considering convenience for encoding, we focus on the distance profiles with
respect to cyclic subcode chains (DPCs) of cyclic codes over $GF(q)$ with
length $n$ such that $\mbox{gcd}(n,q) = 1$. In this paper the optimum DPCs and
the corresponding optimum cyclic subcode chains are investigated on the
punctured second-order Reed-Muller code $\mathcal{RM}(2,m)^*$ for increasing
message length, where two standards on the optimums are studied according to
the rhythm of increase.
