Codes With Locality in the Rank and Subspace Metrics

Kadhe, Swanand; Rouayheb, Salim El; Duursma, Iwan; Sprintson, Alex

doi:10.1109/tit.2019.2915539

Cited by 5 publications

(11 citation statements)

References 65 publications

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“…, 5)) over GF(4) such that C 0 is a [5,2,4] doubly extended RS over GF(4) and V 0 is a [17,8,10] doubly extended RS code over GF (16). This one is then an [85, 16,40] code over GF(4) whose minimum distance d = 40, according to Lemma 19, meets bound (2).…”

Section: Eii Codes Achieving Bound (2) and Comparison With Other mentioning

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: Eii Codes Achieving Bound (2) and Comparison With Other mentioning

confidence: 99%

“…Certainly, this list is not complete and several results are common to different approaches. There is also recent work on LRC codes with the rank metric [40].…”

Section: Introductionmentioning

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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“…The work [11] focuses on codes with locality for the rank metric, allowing to locally correct full columns, whereas a more general case is considered in [13].…”

Section: Conclusion and Open Problemsmentioning

confidence: 99%

“…Since then, codes in the cover metric with other correcting features (e.g., probabilistic, list-decoding, etc.) were subsequently studied [1,11,13,23,24,25].…”

Section: Introductionmentioning

confidence: 99%

Multilayer crisscross error and erasure correction

Martínez-Peñas¹

2022

Preprint

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In this work, multilayer crisscross error and erasures are considered, which affect entire rows and columns in the matrices of a list of matrices. To measure such errors and erasures, the multi-cover metric is introduced. Several bounds are derived, including a Singleton bound, and maximum multi-cover distance (MMCD) codes are defined as those attaining it. Duality, puncturing and shortening of linear MMCD codes are studied. It is shown that the dual of a linear MMCD code is not necessarily MMCD, and those satisfying this duality condition are defined as dually MMCD codes. Finally, some constructions of codes in the multi-cover metric are given, including dually MMCD codes, together with efficient decoding algorithms for them.

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“…The idea of the proof of this statement is similar to the corresponding proof in the Hamming metric in (Tamo and Barg, 2014) and based on determining a polynomial that is constant on all elements of P j for each j ∈ [r + ρ − 1]. For sake of brevity, we omit the proof of this theorem and refer the interested reader to (Kadhe et al, 2019).…”

Section: Codes With Locality In the Rank Metricmentioning

confidence: 99%

Rank-Metric Codes and Their Applications

Bartz¹,

Holzbaur²,

Liu³

et al. 2022

Preprint

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The rank metric measures the distance between two matrices by the rank of their difference. Codes designed for the rank metric have attracted considerable attention in recent years, reinforced by network coding and further motivated by a variety of applications. In code-based cryptography, the hardness of the corresponding generic decoding problem can lead to systems with reduced public-key size. In distributed data storage, codes in the rank metric have been used repeatedly to construct codes with locality, and in coded caching, they have been employed for the placement of coded symbols. This survey gives a general introduction to rank-metric codes, explains their most important applications, and highlights their relevance to these areas of research.Codes in the rank metric have drawn increasing interest in recent years due to their application to cryptography, distributed storage and network coding. The survey by Gorla and Ravagnani (2018) mainly focuses on combinatorial properties of rank-metric codes, while the one by Sheekey (2019a) considers MRD codes and their properties. In particular, Gorla and Ravagnani (2018) and Gorla ( 2019) treat (amongst others): isometries, anticodes, duality, MacWilliams identities, generalized weights in the rank metric. We will therefore not considers these topics in this survey. An English version of the textbook by Gabidulin (2021) has been published recently which contains a collection of Gabidulin's results in the area of rank-metric codes. Our survey deals shortly with properties of rank-metric codes, but the main focus is on their decoding and their applications to code-based cryptography, storage, and network coding.In this chapter, we give an introduction to rank-metric codes, their properties and decoding. In Section 2.1, we provide the basic notation used in this survey. Section 2.2 defines linearized polynomials and recalls some basic properties. In Section 2.3, we define the rank metric and state bounds on the cardinality of rank-metric codes, i.e., spherepacking, Gilbert-Varshamov, and Singleton-like bounds. Section 2.4, 2.5 and 2.6 provide the weight distribution of rank metric codes, constant-rank codes and covering property of rank-metric codes, respectively. Section 2.7 defines Gabidulin codes, proves their minimum rank distance and gives generator and parity-check matrices. In Section 2.8, we describe syndrome-based decoding of Gabidulin codes, i.e., we prove the key equation, show how to solve it and how to reconstruct the final error. We also outline error-erasure decoding and summarize known fast decoders. In Section 2.9, we discuss results on list decoding of Gabidulin codes. Further, we introduce interleaved Gabidulin codes (Section 2.10), folded Gabidulin codes (Section 2.11) and summarize further classes of MRD codes (Section 2.13).

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Codes With Locality in the Rank and Subspace Metrics

Cited by 5 publications

References 65 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

Multilayer crisscross error and erasure correction

Rank-Metric Codes and Their Applications

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