Two-dimensional interleaving using the set partitioning technique

Almeida, Celso de; Palazzo, Reginaldo

doi:10.1109/isit.1994.395120

Cited by 12 publications

(13 citation statements)

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“…Most 2-D burst error correcting codes that have been studied in the literature correct only error bursts of given rectangular [1,5,6,10,12,15,16,21] or other prescribed shapes [2,7,11,20]. See also [17,18] for related interleaving schemes on tori, paths, and cycles, and [4,9,19] for 2-D interleaving schemes for multiple random error correction codes.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

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Optimal interleaving schemes for correcting two-dimensional cluster errors

Golomb

2007

Discrete Applied Mathematics

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We present an elementary theory of optimal interleaving schemes for correcting cluster errors in two-dimensional digital data. It is assumed that each data page contains a fixed number of, say n, codewords with each codeword consisting of m code symbols and capable of correcting a single random error (or erasure). The goal is to interleave the codewords in the m × n array such that different symbols from each codeword are separated as much as possible, and consequently, an arbitrary error burst with size up to t can be corrected for the largest possible value of t. We show that, for any given m, n, the maximum possible interleaving distance, or equivalently, the largest size of correctable error bursts in an m × n array, is given by t = √ 2n if n m 2 /2 , and t = m + (n − m 2 /2 )/m if n m 2 /2 . Furthermore, we develop a simple cyclic shifting algorithm that can provide a systematic construction of an m × n optimal interleaving array for arbitrary m and n. This extends important earlier work on the complementary problem of constructing interleaving arrays that, given the burst size t, minimize the interleaving degree, that is, the number of different codewords in a 2-D (or 3-D) array such that any error burst with given size t can be corrected. Our interleaving scheme thus provides the maximum burst error correcting power without requiring prior knowledge of the size or shape of an error burst.

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Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

“…Fig. 7(b) shows the 5-interleaved 4 × 12 array generated by the procedures in Lemma 4.2 with = (0, 5,9,2). Note that one can further apply Lemma 4.2 to Fig.…”

Section: Proofmentioning

confidence: 99%

Optimal interleaving schemes for correcting two-dimensional cluster errors

Golomb

2007

Discrete Applied Mathematics

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“…Group C: [1,4] , [2,4] , [3,4] is assigned the integer 'j', (ii) there exist two green vertices that are assigned integers 'i' and 'j' respectively such that there is no green vertex between them. (3) A pair [i, j] belongs to group C if and only if one of the following two conditions is satisfied: (i) at least one green vertex is assigned the integer 'i' and no green vertex is assigned the integer 'j', (ii) at least one green vertex is assigned the integer 'j' and no green vertex is assigned the integer 'i'.…”

Section: Proof Of Lemmamentioning

confidence: 99%

“…The material in this paper was presented in part at the 7th International Symposium on Communication Theory and Applications, Ambleside, Lake District, UK, July 13 -18, 2003 interleaving scheme includes [10], [12], [15], [16] and [17], where the underlying graphs on which integers are interleaved include tori, arrays and circulant graphs. In [1], [2] and [3], codewords are interleaved on arrays to correct burst-errors of rectangular shapes, circular shapes, or arbitrary connected shapes.…”

Section: Introductionmentioning

confidence: 99%

Multicluster Interleaving on Paths and Cycles

Jiang

Bruck

2005

IEEE Trans. Inform. Theory

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Abstract-Interleaving codewords is an important method not only for combatting burst-errors, but also for distributed data retrieval. This paper introduces the concept of Multi-Cluster Interleaving (MCI), a generalization of traditional interleaving problems. MCI problems for paths and cycles are studied. The following problem is solved: how to interleave integers on a path or cycle such that any m (m ≥ 2) non-overlapping clusters of order 2 in the path or cycle have at least 3 distinct integers. We then present a scheme using a 'hierarchical-chain structure' to solve the following more general problem for paths: how to interleave integers on a path such that any m (m ≥ 2) nonoverlapping clusters of order L (L ≥ 2) in the path have at least L + 1 distinct integers. It is shown that the scheme solves the second interleaving problem for paths that are asymptotically as long as the longest path on which an MCI exists, and clearly, for shorter paths as well.Index Terms-Burst error, cluster, cycle, file placement, interleaving, multi-cluster interleaving, path.

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“…This approach was considered e. g. in [8,9]. It has the advantage that known error-correcting codes for the Hamming metric can be used.…”

Section: Introductionmentioning

confidence: 99%

Array Codes Correcting Likely Burst Error Patterns

Breitbach

Bossert

Zyablov

et al. 1999

Trans Emerging Tel Tech

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Abstract. This contribution considers data transmission over channels which are corrupted by two-dimensional bursts of errors. A code is constructed that allows to correct one rectangular burst. Its code rate is close to the generalized Singleton bound, and in the case of binary code symbols it can be implemented efficiently using only shift registers and XOR gates. Based on this code, constructions of codes are studied whose error correcting capabilities are restricted to a subset of rectangular bursts. When only likely error patterns can be corrected, the code rate can be raised while accepting only a small increase in error probability. In particular it will be shown that asymptotically these codes permit data transmission with arbitrarily low error probability at a rate beyond the Generalized Singleton bound.

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Two-dimensional interleaving using the set partitioning technique

Cited by 12 publications

References 0 publications

Optimal interleaving schemes for correcting two-dimensional cluster errors

Optimal interleaving schemes for correcting two-dimensional cluster errors

Multicluster Interleaving on Paths and Cycles

Array Codes Correcting Likely Burst Error Patterns

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