The covering radius of the Reed–Muller code <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e112" altimg="si8.svg"><mml:mrow><mml:mi>R</mml:mi><mml:mi>M</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>7</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> is 40

Wang, Qichun

doi:10.1016/j.disc.2019.111625

Cited by 4 publications

(4 citation statements)

References 24 publications

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“…Covering radius of RM (3,7). In 2019, Wang [9] proved that the covering radius of RM (2, 7) is equal to 40. A part of that proof, is based on the classification of B (2,6,6).…”

Section: 2mentioning

confidence: 99%

“…More generally, the classification data of the space B(s, t, m) plays an important role both in coding theory and cryptography. The covering radii of Reed-Muller codes are not generally known and the classification of B(s, t, m) can be used to bound the covering radius of RM (s − 1, m) in RM (t, m) as in the paper [9]. These classifications are also used to study the cryptographic parameters of Boolean functions.…”

Section: Introductionmentioning

confidence: 99%

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Classification of some cosets of the Reed-Muller code

Gillot

Langevin

2023

Cryptogr. Commun.

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This paper presents a descending method to classify Boolean functions in 7 variables under the action of the affine general linear group. The classification determines the number of classes, a set of orbits representatives and a generator set of the stabilizer of each representative. The method consists in the iteration of the classification process of RM (k, m)/RM (r − 1, m) from that of RM (k, m)/RM (r, m). We namely obtain the classifications of RM (4, 7)/RM (2, 7) and of RM (7, 7)/RM (3, 7), from which we deduce some consequences on the covering radius of RM (3, 7) and the classification of near bent functions.

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“…Covering radius of RM (3,7). In 2019, Wang [9] proved that the covering radius of RM (2, 7) is equal to 40. A part of that proof, is based on the classification of B (2,6,6).…”

Section: 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Classification of some cosets of the Reed-Muller code

Gillot

Langevin

2023

Cryptogr. Commun.

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“…Indeed, the covering radii of Reed-Muller codes are not generally known. The classification of B(s, t, m) Date: december 2021. can be used to bound the covering radius of RM (s − 1, m) as in the paper [8]. These classifications are also used to study the cryptographic parameters of Boolean functions.…”

Section: Introductionmentioning

confidence: 99%

Classification of Boolean functions

Gillot¹,

Langevin²

2022

Preprint

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“…The covering radius of RM(1, m) for odd m > 7 is unknown, but has been determined for RM (1,5) [1] and RM(1, 7) [6,3]. In [12], Schatz has found the covering radius of RM (2,6), while recently Wang has established the covering radius of RM(2, 7) [16]. For m > 9, the covering radius of RM(2, m) is still unknown.…”

Section: Introductionmentioning

confidence: 99%

On metric regularity of Reed-Muller codes

Oblaukhov¹

2019

Preprint

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In this work we study metric properties of the well-known family of binary Reed-Muller codes. Let A be an arbitrary subset of the Boolean cube, and A be the metric complement of A -the set of all vectors of the Boolean cube at the maximal possible distance from A. If the metric complement of A coincides with A, then the set A is called a metrically regular set. The problem of investigating metrically regular sets appeared when studying bent functions, which have important applications in cryptography and coding theory and are also one of the earliest examples of a metrically regular set. In this work we describe metric complements and establish metric regularity of the codes RM(0, m) and RM(k, m) for k m − 3. Additionally, metric regularity of the RM(1, 5) code is proved. Combined with previous results by Tokareva N. (2012) concerning duality of affine and bent functions, this proves metric regularity of most Reed-Muller codes with known covering radius. It is conjectured that all Reed-Muller codes are metrically regular.

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The covering radius of the Reed–Muller code RM(2,7) is 40

Cited by 4 publications

References 24 publications

Classification of some cosets of the Reed-Muller code

Classification of some cosets of the Reed-Muller code

Classification of Boolean functions

On metric regularity of Reed-Muller codes

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