  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<= n stations are active at the same time, i.e, are willing to transmit a
message. If in a certain instant at most 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. The measure to optimize is the number of time slots
needed to solve conflicts among all active stations. The fundamental question
is whether this measure decreases linearly with the number d of messages that
can be simultaneously transmitted with success. We give a positive answer to
this question 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 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 this aim, we introduce a new combinatorial
structure that consists in a generalization of Komlos and Greenberg codes.
Constructions of these new codes are obtained via a new generalization of
selectors, whereas the non-existential result is implied by a non-existential
result for a new generalization of the locally thin families.
