Two-dimensional interleaving schemes with repetitions: constructions and bounds

Two-dimensional interleaving schemes with repetitions are considered. These schemes are required for the correction of two-dimensional bursts (or clusters) of errors in applications such as optical recording and holographic storage. We assume that a cluster of errors may have an arbitrary shape, and is characterized solely by its area t. Thus, an interleaving scheme A(t,r) of strength t with r repetitions is an (infinite) array of integers defined by the property that every integer appears no more than r times in any connected component of area t. The problem is to minimize, for a given t and r, the interleaving degree deg A(t,r), which is the total number of distinct integers contained in the array. Optimal interleaving schemes for r=1 (no repetitions) have been devised in earlier work. Here, we consider interleaving schemes for r/spl ges/2. Such schemes reduce the overall redundancy, yet are considerably more difficult to construct and analyze. To this end, we generalize the concept of L/sub 1/-distance and introduce the notions of tristance, quadristance, and more generally r-dispersion. We focus on the special class of interleaving schemes, called lattice interleavers, that is akin to the class of linear codes in coding theory. We construct efficient lattice interleavers for r=2,3,4 and some higher values of r. For r=2,3 we show that these lattice interleavers are either optimal for all t or asymptotically optimal for t/spl rarr//spl infin/. We present the results of an extensive computer search that yields the optimal lattice interleavers for r=2,3,4,5,6 and t up to about 1000. Finally, we consider an alternative connectivity model for clusters, where two elements in an array are connected if they are adjacent horizontally, vertically, or diagonally. We establish relations between interleavers for this model and interleavers for the standard horizontal/vertical connectivity model, and show that these models become equivalent for t/spl rarr//spl infin/. We conclude with some conjectures and open problems.