Error Exponent for Multiple Access Channels: Upper Bounds

The problem of bounding the reliability function of a multiple access channel (MAC) is studied. Two new upper bounds on the error exponent of a two-user discrete memoryless (DM)-MAC are derived. The first bound (sphere packing) is an upper bound on the exponent of the average probability of error and is the first bound of this type that is zero outside the capacity region and thus results in a tighter sphere-packing exponent when compared with the tightest known exponent derived by Haroutunian. The second bound (minimum distance) is an upper bound on the exponent of the maximal (as opposed to average) probability of error. To obtain this bound, first, an upper bound on the minimum Bhattacharyya distance between codeword pairs is derived. For a certain class of two-user DM-MACs, an upper bound on the exponent of maximal probability of error is derived as a consequence of the upper bound on the minimum Bhattacharyya distance. We analytically evaluate the sphere packing bound for uniform composition codes for an additive and nonsymmetric channel and show that it is tight near the boundary of the capacity region, i.e., equal to the random coding lower bound.