Integrated interleaving - a novel ECC architecture

Hassner, M.; Abdel-Ghaffar, Khaled; Patel, A. M.; Koetter, R.; Trager, Barry M.

doi:10.1109/20.917615

Cited by 45 publications

(46 citation statements)

References 2 publications

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“…,9,16)) over GF(2) such that C 0 is a[16,7,6] extended BCH code and V 0 is a parity [m, m − 1, 2] code over (GF(2)) 7 (hence m can take any value). According to Lemma 17 and since d(2) opt[32,7] = 12 = 2d , . .…”

mentioning

confidence: 91%

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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confidence: 91%

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

Blaum

2020

IEEE Trans. Inform. Theory

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“…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%

“…Consider the 3-level EII code C(7,(1,1,3,4,7,7)) of Examples 7, 10, 13 and 14. According to Theorem 15, since m = 6, u 0 = 1, u 1 = 3, u 2 = 4, s 0 = 2, s 1 = 1, s 2 = 1 and s 3 = 2 (and hence,ŝ 3 = s 3 = 2,ŝ 2 = s 2 + s 3 = 3, s 1 = s 1 + s 2 + s 3 = 4), by(18), the minimum distance of this code is d = min {(5)(2) ; (4)(4) ; (3)(5)} = 10.✷…”

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confidence: 98%

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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“…For example, for codes of Construction 2, we have the following algorithm, which bears resemblance to the decoding algorithm used for intergrated interleaving schemes [1]. In the follows, all decoding is bounded distance decoding, and the "syndrome constraint" refers to the fact that for all c = (co) ... ,Cm M 1) E C, we have Z sj Q -iQT = 0, where sCiHT is the Hba-syndrome of subline i.…”

Section: Decodingmentioning

confidence: 99%

“…In this case we can try to do something similar. A very related class of codes turns out to be what is known as integrated interleaving [1], which we describe below. Assume that ni = nl, for all i.…”

Section: Code Constructionsmentioning

confidence: 99%

Reliable Memories with Subline Accesses

Han

Lastras-Montao

2007

2007 IEEE International Symposium on Information Theory

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We study memories protected with error control codes, in which the memory's contents are organized in lines which are read and written to in isolation from other lines. In these memories the available redundancy is structured so as to protect individual lines rather than the entire memory as a whole. Often designers wish to read and write only parts of the memory line, as in some instances this leads to various favorable system design tradeoffs, including better power consumption, increased data access concurrency, etc. (alternatively one may say that designers sometimes would prefer smaller line sizes). Nevertheless when designing systems with such subline accesses it is often found that in order to mantain a given level of reliability, the total amount of redundancy allocated in the memory needs to be increased beyond desirable levels. In this work, we initiate a study of the problem of structuring error control codes to allow subline accesses with good tradeoffs between reliability and redundancy. We motivate and explore a setting in which a "double-lookup" protocol is used in conjunction with certain types of two-level codes, whereby error detection is attained in a first level and error correction using the second level is performed whenever errors are detected in the first level. We obtain lower bounds on redundancy for a given level of reliability and offer a code construction that attains this bound for a certain important class of parameters. We also introduce an alternate construction which allows us to find longer codes under restrictions of the Galois Field size used in the codes.

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Integrated interleaving - a novel ECC architecture

Cited by 45 publications

References 2 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

Reliable Memories with Subline Accesses

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