In this paper we consider matrices of special form introduced in [11] and used for the constructing of resilient functions with cryptographically optimal parameters. For such matrices we establish lower bound 1/log₂(√5+1) = 0.5902... for the important ratio t/t+k of its parameters and point out that there exists a sequence of matrices for which the limit of ratio of these parameters is equal to lower bound. By means of these matrices we construct m-resilient n-variable functions with maximum possible nonlinearity 2 ^n-1-2 ^m+1 for m = 0.5902 . . . n+O (log₂ n). This result supersedes the previous record.