Interleaving schemes for multidimensional cluster errors

We present two-dimensional and three-dimensional interleaving techniques for correcting two- and three-dimensional bursts (or clusters) of errors, where a cluster of errors is characterized by its area or volume. Correction of multidimensional error clusters is required in holographic storage, an emerging application of considerable importance. Our main contribution is the construction of efficient two-dimensional and three-dimensional interleaving schemes. The proposed schemes are based on t-interleaved arrays of integers, defined by the property that every connected component of area or volume t consists of distinct integers. In the two-dimensional case, our constructions are optimal: they have the lowest possible interleaving degree. That is, the resulting t-interleaved arrays contain the smallest possible number of distinct integers, hence minimizing the number of codewords required in an interleaving scheme. In general, we observe that the interleaving problem can be interpreted as a graph-coloring problem, and introduce the useful special class of lattice interleavers. We employ a result of Minkowski, dating back to 1904, to establish both upper and lower bounds on the interleaving degree of lattice interleavers in three dimensions. For the case t≡0 mod 6, the upper and lower bounds coincide, and the Minkowski lattice directly yields an optimal lattice interleaver. For t≠0 mod 6, we construct efficient lattice interleavers using approximations of the Minkowski lattice

Interleaving Schemes for Multidimensional Cluster Errors

We present 2 and 3-dimensional interleaving techniques for correcting 2 and 3- dimensional bursts (or clusters) of errors, where a cluster of errors is characterized by its area or volume. A recent application of correction of 2-dimensional clusters appeared in the context of holographic storage. Our main contribution is the construction of efficient 2 and 3-dimensional interleaving schemes. The schemes are based on arrays of integers with the property that every connected component of area or volume t consists of distinct integers (we call these t-interleaved arrays). In the 2-dimensional case, our constructions are optimal in the sense that they contain the smallest possible number of distinct integers, hence minimizing the number of codes required in an interleaving scheme

Interleaving Schemes on Circulant Graphs with Two Offsets

To be added

Multi-Cluster interleaving in linear arrays and rings

Interleaving codewords is an important method not only for combatting burst-errors, but also for flexible data-retrieving. This paper defines the Multi-Cluster Interleaving (MCI) problem, an interleaving problem for parallel data-retrieving. The MCI problems on linear arrays and rings are studied. The following problem is completely solved: how to interleave integers on a linear array or ring such that any m (m greater than or equal to 2) non-overlapping segments of length 2 in the array or ring have at least 3 distinct integers. We then present a scheme using a 'hierarchical-chain structure' to solve the following more general problem for linear arrays: how to interleave integers on a linear array such that any m (m greater than or equal to 2) non-overlapping segments of length L (L greater than or equal to 2) in the array have at least L + 1 distinct integers. It is shown that the scheme using the 'hierarchical-chain structure' solves the second interleaving problem for arrays that are asymptotically as long as the longest array on which an MCI exists, and clearly, for shorter arrays as well

Optimal Interleaving on Tori

We study t-interleaving on two-dimensional tori, which is defined by the property that any connected subgraph with t or fewer vertices in the torus is labelled by all distinct integers. It 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 spherepacking lower bound. We prove the necessary and sufficient conditions for tori that can be perfectly t-interleaved, and present efficient perfect t-interleaving constructions. The most important contribution of this paper is to prove that the t-interleaving numbers of tori large enough in both dimensions, which constitute by far the majority of all existing cases, is at most one more than the sphere-packing lower bound, and to present an optimal and efficient t-interleaving scheme for them. Then we prove some bounds on the t-interleaving numbers for other cases, completing a general picture for the t-interleaving problem on 2-dimensional tori

Optimal Interleaving on Tori

This paper studies $t$-interleaving on two-dimensional tori. Interleaving has applications in distributed data storage and burst error correction, and is closely related to Lee metric codes. A $t$-interleaving of a graph is defined as a vertex coloring in which any connected subgraph of $t$ or fewer vertices has a distinct color at every vertex. We say that a torus can be perfectly t-interleaved if its t-interleaving number (the minimum number of colors needed for a t-interleaving) meets the sphere-packing lower bound, $\lceil t^2/2 \rceil$. We show that a torus is perfectly t-interleavable if and only if its dimensions are both multiples of $\frac{t^2+1}{2}$ (if t is odd) or t (if t is even). The next natural question is how much bigger the t-interleaving number is for those tori that are not perfectly t-interleavable, and the most important contribution of this paper is to find an optimal interleaving for all sufficiently large tori, proving that when a torus is large enough in both dimensions, its t-interleaving number is at most just one more than the sphere-packing lower bound. We also obtain bounds on t-interleaving numbers for the cases where one or both dimensions are not large, thus completing a general characterization of t-interleaving numbers for two-dimensional tori. Each of our upper bounds is accompanied by an efficient t-interleaving scheme that constructively achieves the bound

Weighted frames of exponentials and stable recovery of multidimensional functions from nonuniform Fourier samples

In this paper, we consider the problem of recovering a compactly supported multivariate function from a collection of pointwise samples of its Fourier transform taken nonuniformly. We do this by using the concept of weighted Fourier frames. A seminal result of Beurling shows that sample points give rise to a classical Fourier frame provided they are relatively separated and of sufficient density. However, this result does not allow for arbitrary clustering of sample points, as is often the case in practice. Whilst keeping the density condition sharp and dimension independent, our first result removes the separation condition and shows that density alone suffices. However, this result does not lead to estimates for the frame bounds. A known result of Groechenig provides explicit estimates, but only subject to a density condition that deteriorates linearly with dimension. In our second result we improve these bounds by reducing the dimension dependence. In particular, we provide explicit frame bounds which are dimensionless for functions having compact support contained in a sphere. Next, we demonstrate how our two main results give new insight into a reconstruction algorithm---based on the existing generalized sampling framework---that allows for stable and quasi-optimal reconstruction in any particular basis from a finite collection of samples. Finally, we construct sufficiently dense sampling schemes that are often used in practice---jittered, radial and spiral sampling schemes---and provide several examples illustrating the effectiveness of our approach when tested on these schemes