A Novel Method for Combining Algebraic Decoding and Iterative Processing

Tang, Xiangyu; Koetter, R.

doi:10.1109/isit.2006.261714

Cited by 42 publications

(30 citation statements)

References 3 publications

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“…✷ The conditions c w ∈ C 0 for 0 w m − 1 and (17) are given as the definition of t-level II-RS codes for s t = 0 in [7], [68], [73], and of t-level EII-RS codes in [8]. As we have seen, t-level EII-RS codes are a special case of the more general class of t-level EII-PC codes.…”

Section: Properties Of Eii-pc Codesmentioning

confidence: 99%

“…As we have seen, t-level EII-RS codes are a special case of the more general class of t-level EII-PC codes. Let us point out that in [32], 2-level II-RS codes are called II codes, while in [68], [73], t-level II-RS codes for t > 2 are called Generalized II (GII) codes. Most papers on II codes follow this convention [36], [76], [77].…”

Section: Properties Of Eii-pc Codesmentioning

confidence: 99%

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Extended Integrated Interleaved Codes Over Any Field With Applications to Locally Recoverable Codes

Blaum

2020

IEEE Trans. Inform. Theory

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Integrated Interleaved (II) and Extended Integrated Interleaved (EII) codes are a versatile alternative for Locally Recoverable (LRC) codes, since they require fields of relatively small size. II and EII codes are generally defined over Reed-Solomon type of codes. A new comprehensive definition of EII codes is presented, allowing for EII codes over any field, and in particular, over the binary field GF(2). The traditional definition of II and EII codes is shown to be a special case of the new definition. Improvements over previous constructions of LRC codes, in particular, for binary codes, are given, as well as cases meeting an upper bound on the minimum distance. Properties of the codes are presented as well, in particular, an iterative decoding algorithm on rows and columns generalizing the iterative decoding algorithm of product codes. Two applications are also discussed: one is finding a systematic encoding of EII codes such that the parity symbols have a balanced distribution on rows, and the other is the problem of ordering the symbols of an EII code such that the maximum length of a correctable burst is achieved.Constructions of LRC codes involve different issues and tradeoffs, like the size of the field and optimality criteria. The same is true for EPC codes, of which, as we have seen above, LRC codes are a special case. In particular, one goal is to keep the size of the required finite field small, since operations over a small field have less complexity than ones over a larger field due to the smaller look-up tables involved. For example, Integrated Interleaved (II) codes [32], [68] over GF(q), where q > max{m, n}, were proposed in [7] as LRC codes. The construction in [45], [46] (stair codes) reduces field size when failures are correlated. Extended Integrated Interleaved (EII) codes [8] unify product codes and II codes.As is the case with LRC codes, construction of EPC codes involves optimality issues. For example, LRC codes optimizing the minimum distance were presented in [67]. Except for special cases, II codes are not optimal as LRC codes, but the codes in [67] require a field of size at least mn. The same happens with EII codes: except for special cases, they do not optimize the minimum distance [8].There are stronger criteria for optimization than the minimum distance in LRC codes. For example, PMDS codes [5], [9], [19], [23], [33], [35] satisfy the Maximally Recoverable (MR) property [23], [25]. The definition of the MR property is extended for EPC codes in [25], but it turns out that EPC codes with the MR property are difficult to obtain. For example, in [25] it was proven that an EPC code EP(n, 1; n, 1; 1) (i.e., one vertical and one horizontal parity per column and row and one extra parity) with the MR property requires a field whose size is superlinear on n.The EII codes presented in [8], of which II codes [7], [68], [73], [76], [77] are special cases, in general assume a field GF(q) such that q > max{m, n} and the individual codes are RS type of codes. An exception occurs in [73], where b...

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Section: Properties Of Eii-pc Codesmentioning

confidence: 99%

Section: Properties Of Eii-pc Codesmentioning

confidence: 99%

Extended Integrated Interleaved Codes Over Any Field With Applications to Locally Recoverable Codes

Blaum

2020

IEEE Trans. Inform. Theory

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“…We start by defining EII codes, which unify product codes and II codes. II codes may be interpreted as m × n arrays such that each row belongs in a code C 0 , and certain linear combinations of the rows belong in nested subcodes of C 0 [2], [40], [42], [44]. In addition, we assume that each column is also in a (vertical) code, making the arrays a subcode of a product code.…”

Section: A Definition Of Extended Integrated Interleaved (Eii) Codesmentioning

confidence: 99%

“…Example 2. Assume that s t = 0 in Definition 1, then, in (4), i 1 and C(n, u) is a t-level II code [2], [40], [42]. So, t-level II codes can be viewed as a special case of t-level EII codes.…”

Section: A Definition Of Extended Integrated Interleaved (Eii) Codesmentioning

confidence: 99%

“…In particular, one goal is to keep the size of the required finite field small, since operations over a small field have less complexity than ones over a larger field due to the smaller look-up tables involved. For example, Integrated Interleaved (II) codes [18], [40] over GF(q), where q > max{m, n}, were proposed in [2] as LRC codes (II codes are closely related to Generalized Concatenated Codes [6], [45]). Let us mention also the construction in [26] (STAIR codes), which reduces field size when failures are correlated.…”

Section: Introductionmentioning

confidence: 99%

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Extended Product and Integrated Interleaved Codes

Blaum

Hetzler

2018

IEEE Trans. Inform. Theory

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A new class of codes, Extended Product (EPC) Codes, consisting of a product code with a number of extra parities added, is presented and applications for erasure decoding are discussed. An upper bound on the minimum distance of EPC codes is given, as well as constructions meeting the bound for some relevant cases. A special case of EPC codes, Extended Integrated Interleaved (EII) codes, which naturally unify Integrated Interleaved (II) codes and product codes, is defined and studied in detail. It is shown that EII codes often improve the minimum distance of II codes with the same rate, and they enhance the decoding algorithm by allowing decoding on columns as well as on rows. It is also shown that EII codes allow for encoding II codes with an uniform distribution of the parity symbols.

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Multi-erasure locally recoverable codes over small fields

Huang

Yaakobi

Siegel

2017

2017 55th Annual Allerton Conference on Communication, Control, and Computing (Allerton)

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Erasure codes play an important role in storage systems to prevent data loss. In this work, we study a class of erasure codes called Multi-Erasure Locally Recoverable Codes (ME-LRCs) for storage arrays. Compared to previous related works, we focus on the construction of ME-LRCs over small fields. We first develop upper and lower bounds on the minimum distance of ME-LRCs. Our main contribution is to propose a general construction of ME-LRCs based on generalized tensor product codes, and study their erasure-correcting properties. A decoding algorithm tailored for erasure recovery is given, and correctable erasure patterns are identified. We then prove that our construction yields optimal ME-LRCs with a wide range of code parameters, and present some explicit ME-LRCs over small fields. Finally, we show that generalized integrated interleaving (GII) codes can be treated as a subclass of generalized tensor product codes, thus defining the exact relation between these codes.

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A Novel Method for Combining Algebraic Decoding and Iterative Processing

Cited by 42 publications

References 3 publications

Extended Integrated Interleaved Codes Over Any Field With Applications to Locally Recoverable Codes

Extended Integrated Interleaved Codes Over Any Field With Applications to Locally Recoverable Codes

Extended Product and Integrated Interleaved Codes

Multi-erasure locally recoverable codes over small fields

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