Constructions of Locally Recoverable Codes Which are Optimal

Micheli, Giacomo

doi:10.1109/tit.2019.2939464

Cited by 37 publications

(21 citation statements)

References 17 publications

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“…The genus g f can be bounded solely in terms of deg f , using for example Castelnuovo's inequality. As noticed in [16,Proposition 3.3…”

Section: Monodromy Groups and Totally Split Placesmentioning

confidence: 70%

“…Thanks to the the techniques introduced in [16], given a separable polynomial f ∈ F q [X] such that A(f ) = G(f ), one can obtain an explicit estimate on cardinality of the set…”

Section: Monodromy Groups and Totally Split Placesmentioning

confidence: 99%

“…The machinery we use involves Galois theory, the classification of transitive subgroups of the symmetric group S n up to n = 5, and the theory of function fields, using the results and techniques of [15,16], then further developed in [1,2,4,5,8,9].…”

Section: Introductionmentioning

confidence: 99%

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Optimal Selection for Good Polynomials of Degree up to Five

Dukes¹,

Ferraguti²,

Micheli³

2021

Preprint

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Good polynomials are the fundamental objects in the Tamo-Barg constructions of Locally Recoverable Codes (LRC). In this paper we classify all good polynomials up to degree 5, providing explicit bounds on the maximal number ℓ of sets of size r+1 where a polynomial of degree r+1 is constant, up to r = 4. This directly provides an explicit estimate (up to an error term of O( √ q), with explict constant) for the maximal length and dimension of a Tamo-Barg LRC. Moreover, we explain how to construct good polynomials achieving these bounds. Finally, we provide computational examples to show how close our estimates are to the actual values of ℓ, and we explain how to obtain the best possible good polynomials in degree 5.

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“…The genus g f can be bounded solely in terms of deg f , using for example Castelnuovo's inequality. As noticed in [16,Proposition 3.3…”

Section: Monodromy Groups and Totally Split Placesmentioning

confidence: 70%

“…Thanks to the the techniques introduced in [16], given a separable polynomial f ∈ F q [X] such that A(f ) = G(f ), one can obtain an explicit estimate on cardinality of the set…”

Section: Monodromy Groups and Totally Split Placesmentioning

confidence: 99%

See 1 more Smart Citation

Optimal Selection for Good Polynomials of Degree up to Five

Dukes¹,

Ferraguti²,

Micheli³

2021

Preprint

Self Cite

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“…Specifically speaking, for c ∈ F q , if f (T ) − c splits into n distinct linear factors in F q [T ], then the rational place (t − c) of F q (t) splits completely in F q (x). In 1970, Cohen showed the distribution of f (T ) − c with prescribed factorization as c varies over F q ([1]), and recently Micheli gave a more specific description in [5]. The main result we need is presented in the following lemma.…”

Section: Lemma 2 ([2]mentioning

confidence: 99%

A Function Field Approach Toward Good Polynomials for Further Results on Optimal LRC Codes

Chen,

Mesnager

2021

Preprint

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Because of the recent applications to distributed storage systems, researchers have introduced a new class of block codes, i.e., locally recoverable (LRC) codes. LRC codes can recover information from erasure(s) by accessing a small number of erasure-free code symbols and increasing the efficiency of repair processes in large-scale distributed storage systems. In this context, Tamo and Barg first gave a breakthrough by cleverly introducing a good polynomial notion. Constructing good polynomials for locally recoverable codes achieving Singleton-type bound (called optimal codes) is challenging and has attracted significant attention in recent years. This article aims to increase our knowledge on good polynomials for optimal LRC codes. Using tools from algebraic function fields and Galois theory, we continue investigating those polynomials and studying them by developing the Galois theoretical approach initiated by Micheli in 2019. Specifically, we push further the study of a crucial parameter G(f ) (of a given polynomial f ), which measures how much a polynomial is "good" in the sense of LRC codes. We provide some characterizations of polynomials with minimal Galois groups and prove some properties of finite fields where polynomials exist with a specific size of Galois groups. We also present some explicit shapes of polynomials with small Galois groups. For some particular polynomials f , we give the exact formula of G(f ).

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“…Afterwards, we prove in Section 4 several necessary conditions for a polynomial to be exceptional L-q t -partially scattered, classifying for instance those of index at most 1. This is done by means of tools already exploited in the literature in connection with the exceptional scatteredness, such as algebraic curves and function fields over finite fields, also exploiting a method due to G. Micheli in [31,32]. Interestingly, such connections can be generalized to exceptional L-q t -partial scatteredness.…”

Section: Introductionmentioning

confidence: 99%

Exceptional scattered polynomials

Bartoli

Zhou

2018

Journal of Algebra

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Let f be an Fq-linear function over F q n . If the Fq-subspace U = {(x q t , f (x)) : x ∈ F q n } defines a maximum scattered linear set, then we call f a scattered polynomial of index t. As these polynomials appear to be very rare, it is natural to look for some classification of them. We say a function f is an exceptional scattered polynomial of index t if the subspace U associated with f defines a maximum scattered linear set in PG(1, q mn ) for infinitely many m. Our main results are the complete classifications of exceptional scattered monic polynomials of index 0 (for q > 5) and of index 1. The strategy applied here is to convert the original question into a special type of algebraic curves and then to use the intersection theory and the Hasse-Weil theorem to derive contradictions.

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Constructions of Locally Recoverable Codes Which are Optimal

Cited by 37 publications

References 17 publications

Optimal Selection for Good Polynomials of Degree up to Five

Optimal Selection for Good Polynomials of Degree up to Five

A Function Field Approach Toward Good Polynomials for Further Results on Optimal LRC Codes

Exceptional scattered polynomials

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