Finite two-user interference channel and strong-interference regime

A finite two-user interference channel is the two-sender, two-receiver memoryless channel with finite nonempty input alphabets ${\mathcal{X}}_1,{\mathcal{X}}_2$, finite nonempty output alphabets ${\mathcal{Y}}_1,{\mathcal{Y}}_2$, and transition law $W(y_1,y_2\mathrm{\mid }{x}_1,x_2)$. Independent uniform messages $M_1,M_2$ on finite nonempty sets are encoded separately as $X_1^n=f_1(M_1)$ and $X_2^n=f_2(M_2)$; receiver $i$ estimates $M_i$ from $Y_i^n$. The strong-interference regime means that, for every product law $P_{X_1}{P}_{X_2}$, $I(X_1;Y_1\mathrm{\mid }{X}_2)\le I(X_1;Y_2\mathrm{\mid }{X}_2),I(X_2;Y_2\mathrm{\mid }{X}_1)\le I(X_2;Y_1\mathrm{\mid }{X}_1)\mathrm{.}$ The inequalities are evaluated under the stated product law and the channel. They are universal hypotheses over input laws, not just inequalities at a chosen optimizing distribution.