We have developed an asymmetric ferromagnetic NOT gate and shift register optimized on a square grid. This gives rise to a two-dimensional storage scheme built up by tessellating an elementary data unit, which is scalable down to very narrow wire widths. The areal footprint of each storage unit is 15F2, where F is the minimum feature size. We experimentally demonstrate NOT operations across a chain of three gates made from Permalloy with F = 60 nm, and present a functional 15-gate, multichannel shift register with electrical injection, and optical readout.