Incorrigible set distributions and unsuccessful decoding probability of linear codes

Abstract

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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> } <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i=0</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> 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].