In this paper, we investigate the relative power of several conjectures that attracted recently lot of interest. We establish a connection between the Network Coding Conjecture (NCC) of Li and Li and several data structure like problems such as non-adaptive function inversion of Hellman and the well-studied problem of polynomial evaluation and interpolation. In turn these data structure problems imply super-linear circuit lower bounds for explicit functions such as integer sorting and multi-point polynomial evaluation.

We build boolean circuits of size $O(nm^2)$ and depth $O(\log(n) + m \log(m))$ for sorting $n$ integers each of $m$-bits. We build also circuits that sort $n$ integers each of $m$-bits according to their first $k$ bits that are of size $O(nmk(1 + \log^*(n) - \log^*(m)))$ and depth $O(\log^{3}(n))$. This improves on the result of Asharov et al. arXiv:2010.09884 and resolves some of their open questions.