A k-packing coloring of a graph G = (V, E) is a mapping from V to {1, ..., k} such that any pair of vertices u, v that receive the same color i must be at distance greater than i in G. Arguably the most fundamental problem regarding packing colorings is to determine the packing chromatic number of the infinite square grid. A sequence of previous works has proved this number to be between 13 and 15. Our work improves the lower bound to 14. Moreover, we present a new encoding that is asymptotically more compact than the previously used ones.