The index coding problem involves a sender with K messages to be transmitted across a broadcast channel, and a set of receivers each of which demands a subset of the K messages while having prior knowledge of a different subset as side information. We consider the specific instance of noisy index coding where the broadcast channel is Gaussian and every receiver demands all the messages from the source. We construct lattice index codes for this channel by encoding the K messages individually using K modulo lattice constellations and transmitting their sum modulo a shaping lattice. We introduce a design metric called side information gain that measures the advantage of a code in utilizing the side information at the receivers, and hence its quality as an index code. Based on the Chinese remainder theorem, we then construct lattice index codes for the Gaussian broadcast channel. Among all lattice index codes constructed using any densest lattice of a given dimension, our codes achieve the maximum side information gain. © 2015 IEEE.