Two-dimensional interleaving schemes with repetitions: constructions and bounds

The most common approach for dealing with one-dimensional error bursts is interleaving. While the optimal one-dimensional interleaving schemes, both with and without repetitions, are straightforward, in two dimensions, it is not at all obvious how to interleave a minimal number of codewords so that any burst of size up to t can be corrected. Most two-dimensional burst-correcting codes that have been studied in the literature so far correct error bursts of a given rectangular shape. In this work, we assume that a cluster of errors can have an arbitrary shape, as long as it maintains horizontal/vertical connectivity. Given this notion of a cluster, one may define a two-dimensional interleaving scheme A(t,r) of strength t with r repetitions as an infinite array of integers characterized by the property that every integer appears at most r times in any cluster of size t. We introduce the notion of r-dispersion that turns out to be crucial in the design of two-dimensional interleaving schemes with repetitions. The r-dispersion may be thought of as a generalization of the L₁-distance to a quantity that reflects a property of r points for r⩾2