A New Sphere-Packing Bound for Maximal Error Exponent for   Multiple-Access Channels
  In this work, a new lower bound for the maximal error probability of a two-user discrete memoryless (DM) multiple-access channel (MAC) is derived. This is the first bound of this type that explicitly imposes independence of the users' input distributions (conditioned on the time-sharing auxiliary variable) and thus results in a tighter sphere-packing exponent when compared to the tightest known exponent derived by Haroutunian. 
Ali Nazari, Sandeep Pradhan, Achilleas Anastasopoulos