Optimal Interleaving on Tori

This paper studies t-interleaving on two-dimensional tori, which is defined by the property that every connected subgraph of order t in the torus is labelled by t distinct integers. This is the first time that the t-interleaving problem is solved for graphs of modular structures. t-interleaving on tori has applications in distributed data storage and burst error correction, and is closely related to Lee metric codes. We say that a torus can be perfectly t-interleaved if its t-interleaving number — the minimum number of distinct integers needed to t-interleave the torus — meets the sphere-packing lower bound. We prove the necessary and sufficient conditions for tori that can be perfectly tinterleaved, and present efficient perfect t-interleaving constructions. The most important contribution of this paper is to prove that when a torus is large enough in both dimensions, its t-interleaving number is at most one more than the sphere-packing lower bound, and to present an optimal and efficient t-interleaving scheme for such tori. Then we prove bounds for the t-interleaving numbers of the remaining cases, completing a general characterization of the t-interleaving problem on 2-dimensional tori.