The standard Cantor pairing function $J:\Bbb N \times \Bbb N \to \Bbb N$ which enumerates antidiagonals is efficient in the sense that it assigns $(m,n)$ in increasing order by $m+n$ (so that this can be viewed as the "complexity metric" that is minimized by this pairing function). But it performs very poorly when one input is much larger than the other; for example $J(0,n)=O(n^2)$, which means that using the pairing function in an unbalanced way for higher tuples, i.e. $J(x_1, J(x_2,\dots, J(x_{n-1},x_n)))$ leads to exponential growth $O(k^{2n})$ when $x_n=k$ and the others are fixed to, say, $0$.
Information entropy analysis suggests that to store two numbers $m,n$ should require approximately $\log_2 m+\log_2n$ bits to store, and a pairing function with this property would show linear growth, not exponential, in unbalanced pairing situations.
Is there a pairing function such that $\ell(J(m,n))\le\ell(m)+\ell(n)+O(\log\log(m+n))$, where $\ell(x)=\lfloor\log_2x\rfloor+1\approx \log_2 x$? (It need not be bijective, just injective, if that makes it easier.) One encoding strategy is to read $\bar m,\bar n\in\{0,1\}^*$ as binary strings, and then form the expression $\bar m\,\$\,\bar n$ as a string in the alphabet $\{0,1,\$\}$, and encode in ternary. But this yields $\ell(J(m,n))=\log_23(\ell(m)+\ell(n)+1)$ which is off by a factor $\log_23$ from optimal.
(Bonus points if you can give an optimal pairing function that uses only standard algebraic operations without resorting to bit tricks like this.)
Encode the number $k=\ell(m)$ of bits in $m$ as follows: Encode one bit from $k$, then one bit to signal whether there are more; if so, then two more bits from $k$, then again one to signal whether there are more; if so, then four more bits from $k$, etc., using $O(\ell(k))$ and thus $O(\log\log m)$ bits; then append $m$ and $n$.