On the bounds and achievability about the ODPC of $\mathcal{GRM}(2,m)^*$ over prime field for increasing message length

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.