The Information Complexity of Hamming Distance

The Hamming distance function Ham_{n,d} returns 1 on all pairs of inputs x and y that differ in at most d coordinates and returns 0 otherwise. We initiate the study of the information complexity of the Hamming distance function. We give a new optimal lower bound for the information complexity of the Ham_{n,d} function in the small-error regime where the protocol is required to err with probability at most epsilon < d/n. We also give a new conditional lower bound for the information complexity of Ham_{n,d} that is optimal in all regimes. These results imply the first new lower bounds on the communication complexity of the Hamming distance function for the shared randomness two-way communication model since Pang and El-Gamal (1986). These results also imply new lower bounds in the areas of property testing and parity decision tree complexity