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We consider the Conflict Resolution Problem in the context of a multiple-access system in which several stations can transmit their messages simultaneously to the channel. We assume that there are n stations and that at most k, k <= n, stations are active at the same time, i.e, are willing to transmit a message over the channel. If in a certain instant at most d, d <= k, active stations transmit to the channel then their messages are successfully transmitted, whereas if more than d active stations transmit simultaneously then their messages are lost. In this latter case we say that a conflict occurs. The present paper investigates non-adaptive conflict resolution algorithms working under the assumption that active stations receive a feedback from the channel that informs them on whether their messages have been successfully transmitted. If a station becomes aware that its message has been correctly sent over the channel then it becomes immediately inactive, that is, stops transmitting. The measure to optimize is the number of time slots needed to solve conflicts among all active stations. The fundamental question is how much this measure decreases with the number d of messages that can be simultaneously transmitted with success. In this paper we prove that it is possible to achieve a speedup linear in d by providing a conflict resolution algorithm that uses a 1/d ratio of the number of time slots used by the optimal conflict resolution algorithm for the particular case d = 1. Moreover, we derive a lower bound on the number of time slots needed to solve conflicts non-adaptively which is within a log(k/d) factor from the upper bound. To the aim of proving these results, we introduce a new combinatorial structure that consists in a generalization of Komlós and Greenberg codes. Constructions of these new codes are obtained via a new kind of selectors, whereas the non-existential result is implied by a non-existential result for a new generalization of the locally thin families. We believe that the combinatorial structures introduced in this paper and the related results may be of independent interest.