When does the modular distance induce a metric in the binary case? (Corresp.)

The modular distance induces a metric if and only if the nonadjacent form of the modulusMhas one of the following forms:1) 2^{n}+2^{n-2} pm 2^{i}, wheren-igeq 4; 2) 2^{n} - 2^{j} pm 2^{i}, where2 leq n -j leq 5andj-igeq 2; 3) 2^{n} pm 2^{j}, wheren -j geq 2; 4) 2^{n}.