A system of linear equations is Sidorenko over F_p if any subset of F_p^n contains at least as many solutions to it as a random set of the same density, asymptotically as n->infty. A system of linear equations is common over F_p if any 2-coloring of F_p^n gives at least as many monochromatic solutions to it as a random 2-coloring, asymptotically as n->infty.

It is an open question which linear systems are common or Sidorenko. For a single linear equation, a complete characterization of common and Sidorenko equations was given by Fox-Pham-Zhao; for linear systems that contain multiple equations, progress has been made by Versteegen, Kamčev-Liebenau-Morrison and Altman. In joint work with Anqi Li and Yufei Zhao, we:

- determine commonness of all 2*5 linear systems up to a small number of cases;

- answering a question of Kamčev-Liebenau-Morrison, show that all 2*k linear systems with k even and girth (length of the shortest equation) k-1 are uncommon.