New bounds on the field size for maximally recoverable codes instantiating grid-like topologies

Kong, Xiangliang; Ma, Jingxue; Ge, Gennian

doi:10.1007/s10801-021-01013-1

Cited by 7 publications

(6 citation statements)

References 42 publications

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“…We also show the following upper bound on field size for MR tensor codes, generalizing [KMG21] where they obtained such a result for a = 1. Theorem 1.7.…”

Section: Our Contributionssupporting

confidence: 52%

“…Lower bounds: Prior to our work there are no general lower bounds on the field size required for MR tensor codes. In the special case of (m = 4, n, a = 1, b = 2)-MR tensor code, [KMG21] prove a quadratic lower bound on the field size, i.e., q = Ω(n 2 ). For codes with topology given by T n×n (a = 1, b = 1, h = 1), [GHK + 17] prove a lower bound of exp Ω log(n) 2 .…”

Section: Previous Workmentioning

confidence: 89%

“…This is the first general lower bound on the field size required for MR tensor codes. Prior to our work, only a quadratic lower bound was known in the special case of (m = 4, n, a = 1, b = 2) [KMG21], but unfortunately we cannot recover this bound from our general result.…”

Section: Our Contributionsmentioning

confidence: 95%

“…Constructions: If we instantiate the row code and column code with random codes over fields of size q ≫ (an + bm − ab) • mn an+bm−ab , by Schwartz-Zippel lemma and union bound, we can conclude that with high probability the tensor code will be MR. By doing a more careful union bound, [KMG21] show that there exists (m, n, a = 1, b)-MR tensor codes over fields of size q = O m,b n b(m−1) . In some special cases, (m = 4, n, a = 1, b = 2) and (m = 3, n, a = 1, b = 3), [KMG21] proved the existence of MR tensor codes over fields of size q = O(n 5 ).…”

Section: Previous Workmentioning

confidence: 99%

“…A computer search can also find a number of other counterexamples, such as MDS(8, 4, 4) codes whose duals are not MDS(4). Another way to see the failure of duality of MDS(4) is by the lower bound of [KMG21] where they proved that (m = 4, n, a = 1, b = 2)-MR tensor codes require fields of size Ω(n 2 ). Constructing (m = 4, n, a = 1, b = 2)-MR tensor codes is equivalent to constructing (n, n−2)−MDS(4) code by Theorem 1.4.…”

Section: B3 Counterexample To Mds(ℓ) Duality For ℓmentioning

confidence: 99%

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Lower Bounds for Maximally Recoverable Tensor Code and Higher Order MDS Codes

Brakensiek¹,

Gopi²,

Makam³

2021

Preprint

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An (m, n, a, b)-tensor code consists of m × n matrices whose columns satisfy 'a' parity checks and rows satisfy 'b' parity checks (i.e., a tensor code is the tensor product of a column code and row code). Tensor codes are useful in distributed storage because a single erasure can be corrected quickly either by reading its row or column. Maximally Recoverable (MR) Tensor Codes, introduced by Gopalan et al. [GHK + 17], are tensor codes which can correct every erasure pattern that is information theoretically possible to correct. The main questions about MR Tensor Codes are characterizing which erasure patterns are correctable and obtaining explicit constructions over small fields.In this paper, we study the important special case when a = 1, i.e., the columns satisfy a single parity check equation. We introduce the notion of higher order MDS codes (MDS(ℓ) codes) which is an interesting generalization of the well-known MDS codes, where ℓ captures the order of genericity of points in a low-dimensional space. We then prove that a tensor code with a = 1 is MR iff the row code is an MDS(m) code. We then show that MDS(m) codes satisfy some weak duality. Using this characterization and duality, we prove that (m, n, a = 1, b)-MR tensor codes require fields of size q = Ω m,b (n min{b,m}−1 ). Our lower bound also extends to the setting of a > 1. We also give a deterministic polynomial time algorithm to check if a given erasure pattern is correctable by the MR tensor code (when a = 1).

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“…We also show the following upper bound on field size for MR tensor codes, generalizing [KMG21] where they obtained such a result for a = 1. Theorem 1.7.…”

Section: Our Contributionssupporting

confidence: 52%

Section: Previous Workmentioning

confidence: 89%

Section: Our Contributionsmentioning

confidence: 95%

Section: Previous Workmentioning

confidence: 99%

Section: B3 Counterexample To Mds(ℓ) Duality For ℓmentioning

confidence: 99%

See 3 more Smart Citations