부산대학교 도서관

This paper is written in an erudite manner. It is absolutely true that it has been an open problem for many years to find the covering radius of the third-order Reed-Muller code of length 128. The authors give a sufficient and necessary condition for the covering radius of $RM(3,7)$ to be equal to 22, which is known as the best upper bound. To be precise, they give a bound on the covering radius of $RM(3,7)$ in the set of 2-resilient Boolean functions, modifying the bound given by Borissov. The introduction explains this very clearly. In the preliminaries, the authors define the covering radius and $n$-variable Boolean function very clearly. They give many important observations which are used to find the sufficient and necessary conditions. In those conditions on the covering radius of the Reed-Muller code $RM(3,7)$, all lemmas, propositions and their proofs are very well explained, step by step. The lemmas on the covering radius of $RM(3,7)$ in $RM(4,7)$ are the best results in this paper. It is a very valuable result that the covering radius of $RM(3,7)$ in $RM(4,7)$ is 20. I would like to have seen some applicable examples or data for the results proved. The research in this paper is very useful from a cryptographic point of view.