On the Optimum Cyclic Subcode Chains of ${\cal RM}(2, m)^{\ast}$ for Increasing Message Length

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, where one example is in the design of transport format combination indicators (TFCIs) in CDMA. 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 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 RM(2, m)* for increasing message length, where two standards on the optimums are studied according to the rhythm of increase. Ignoring the dimension profile, the device will coincide with that of TFCI.