On the covering radius of the third order Reed–Muller code RM(3, 7)

Abstract

The covering radius of the third order Reed–Muller code of length 128 has been an open problem for many years. The best upper bound of it is known to be 22. In this paper, we give a sufficient and necessary condition for the covering radius of RM(3, 7) to be equal to 22. Using this condition, we prove that the covering radius of RM(3, 7) in RM(4, 7) is 20. Therefore, if the third-order nonlinearity of a 7-variable Boolean function is greater than 20, then its algebraic degree is at least 5. As a corollary, we conclude that the covering radius of RM(3, 7) in the set of 2-resilient Boolean functions is at most 20 which improves the bound given by Borissov et al. (IEEE Trans Inf Theory 51:1182–1189, 2005).