We propose a method for the computation of entropy decrease in C-space. These estimates are then used to evaluate candidate exploratory trajectories in the context of autonomous mobile robot mapping. The method evaluates both map and path entropy reduction and uses such estimates to compute trajectories that maximize coverage whilst min- imizing localization uncertainty, hence reducing map error. Very efficient kernel convolution mechanisms are used to evaluate entropy reduction at each sensor ray, and for each possible robot position and orientation, taking frontiers and obstacles into account. In contrast to most other exploration methods that evaluate entropy reduction at a small number of discrete robot configurations, we do it densely for the entire C-space. The computation of such dense entropy reduction maps opens the window to new exploratory strategies. In this paper we present two such strategies. In the first one we drive exploration through a gradient descent on the entropy decrease field. The second strategy chooses maximal entropy reduction configurations as candidate exploration goals, and plans paths to them using RRT*. Both methods use PoseSLAM as their estimation backbone, and are tested and compared with classical frontier-based exploration in simulations using common publicly available datasets.