Euclidean division method is one example of mapping finite sequences $s^K_i$ of natural integers to rational numbers $x_s$, satisfying, if my understanding is correct, a condition of uniqueness both ways.
Here, there's a variation of the problem in a multi-dimensional space. Suppose there are N finite sequences of non-negative integers of the same length l, with only one non-zero element at each position $k \le l$: if $s^n_k > 0$ then $s^j_k = 0 \; \forall \; 1 \le j \le N, j \ne n$.
Is a method known of mapping of such sequences to N-dimensional rational numbers, also with the condition of uniqueness of mapping both ways?