Optimal t-interleaving on tori

The number of integers needed to t-interleave a 2-dimensional torus has a sphere-packing lower bound. We present the necessary and sufficient conditions for tori to meet that lower bound. We prove that for tori sufficiently large in both dimensions, their t-interleaving numbers exceed the lower bound by at most 1. We then show upper bounds on t-interleaving numbers for other cases, completing a general picture for the problem of t-interleaving on 2-dimensional tori. Efficient t-interleaving algorithms are also presented.