Improved cardinality bounds on the auxiliary random variables in Marton´s inner bound

Marton´s region is the best known inner bound for a general discrete memoryless broadcast channel. We establish improved bounds on the cardinalities of the auxiliary random variables. We combine the perturbation technique along with a representation using concave envelopes to achieve this improvement. As a corollary of this result, we show that a randomized time division strategy achieves the entire Marton´s region for binary input broadcast channels, extending the previously known result for the sum-rate and validating a previous conjecture due to the same authors.