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In this paper, a new five-step discrete-time zeroing dynamics (DTZD) algorithm, discretized from a continuous-time zeroing dynamics (CTZD) model, is proposed and investigated for online future nonlinear minimization (OFNM), i.e., online discrete-time dynamic nonlinear minimization. For approximating more accurately the first-order derivative and discretizing more effectively the CTZD model, a six-node g-cube discretization (6Ng CD) formula with higher precision is presented to obtain the new five-step DTZD algorithm. Besides, the corresponding theoretical result shows that the proposed five-step DTZD algorithm is with a quartic steady-state error pattern, i.e., O(g4) pattern, with g denoting the sampling gap. Moreover, a general DTZD algorithm is constructed by applying the general linear multistep method, and a specific DTZD algorithm based on the 4th-order Adams-Bashforth method (termed DTZD-AB algorithm for short) is further developed for OFNM. Several numerical experiments are conducted to substantiate the efficacy, accuracy, and superiority of the proposed five-step DTZD algorithm (as well as the DTZD-AB algorithm) for solving the OFNM problem, as compared with the one-step and three-step DTZD algorithms developed and investigated in previous works. [ABSTRACT FROM AUTHOR]