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On the Optimum Cyclic Subcode Chains of $\mathcal{RM}(2,m)^*$ 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. %, 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.