Regularity questions for complex Hadamard matrices

We study the partial Hadamard matrices $H\in M_{M\times N}(\mathbb C)$ which are regular, in the sense that the scalar products between pairs of distinct rows decompose as sums of cycles (rotated sums of roots of unity). The simplest non-trivial case is M=3, and we obtain here several results, notably with a classification at N=7. We discuss as well the potential applications of the M=3 results to various $M=N$ questions.