Incorrigible set distributions and unsuccessful decoding probability of linear codes

On a binary erasure channel (BEC) with erasing probability e, the performance of a binary linear code is determined by the incorrigible sets of the code. The incorrigible set distribution (ISD) {I_i}_i=0ⁿ enumerates the number of incorrigible sets with size i of the code. The probability of unsuccessful decoding under optimal decoding for the code could be formulated by the ISD and ∈. In this paper, we determine the ISDs for the Simplex codes, the Hamming codes, the first order Reed-Muller codes, and the extended Hamming codes, which are some Reed-Muller codes or their shortening or puncturing versions. Then, we show that the probability of unsuccessful decoding under optimal decoding for any binary linear code is monotonously non-decreasing on ∈ in the interval [0,1].