  In this work we show how an improved lower bound to the error exponent of the
memoryless multiple-access (MAC) channel is attained via the use of linear
codes, thus demonstrating that structure can be beneficial even in cases where
there is no capacity gain. We show that if the MAC channel is modulo-additive,
then any error probability, and hence any error exponent, achievable by a
linear code for the corresponding single-user channel, is also achievable for
the MAC channel. Specifically, for an alphabet of prime cardinality, where
linear codes achieve the best known exponents in the single-user setting and
the optimal exponent above the critical rate, this performance carries over to
the MAC setting. At least at low rates, where expurgation is needed, our
approach strictly improves performance over previous results, where expurgation
was used at most for one of the users. Even when the MAC channel is not
additive, it may be transformed into such a channel. While the transformation
is lossy, we show that the distributed structure gain in some "nearly additive"
cases outweighs the loss, and thus the error exponent can improve upon the best
known error exponent for these cases as well. Finally we apply a similar
approach to the Gaussian MAC channel. We obtain an improvement over the best
known achievable exponent, given by Gallager, for certain rate pairs, using
lattice codes which satisfy a nesting condition.
