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 modulus
M
has one of the following forms:
1) 2^{n}+2^{n-2} \pm 2^{i}
, where
n-i\geq 4; 2) 2^{n} - 2^{j} \pm 2^{i}
, where
2 \leq n -j \leq 5
and
j-i\geq 2; 3) 2^{n} \pm 2^{j}
, where
n -j \geq 2; 4) 2^{n}
.