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

TL;DR

This paper investigates the optimal cyclic subcode chains of punctured second-order Reed-Muller codes to enhance distance profiles for variable message lengths, aiming to improve encoding efficiency and error correction.

Contribution

It introduces a method to identify the optimum distance profile chains of cyclic subcodes in punctured Reed-Muller codes for increasing message lengths.

Findings

Identified optimal cyclic subcode chains for $ ext{RM}(2,m)^*$.

Analyzed distance profiles with respect to message length increase.

Provided standards for optimality based on growth rhythm.

Abstract

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.