On the covering radius of the third order Reed–Muller code RM(3, 7)

Wang, Qichun; Tan, Chik How; Prabowo, Theo Fanuela

doi:10.1007/s10623-017-0329-5

Cited by 10 publications

(6 citation statements)

References 18 publications

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“…These computations give a new (and quickly verifiable) proof of the result of Wang, Tan, and Prabowo [25] that the covering radius of RM (3,7) in RM(4, 7) is 20.…”

Section: Polynomials In 7 or Variablesmentioning

confidence: 58%

“…It is known that ρ(2, 1, 5) = 8 [13] and ρ(2, 2, 6) = 16 [1]. Reference [1] also gives the bounds 16 ≤ ρ(2, 3, 7) ≤ 22 and 16 ≤ ρ(2, 4, 8), while [25] gives ρ(2, 3, 7) ≤ 20.…”

Section: Parametersmentioning

confidence: 99%

“…Hou [12] later gave a noncomputer proof of this result. Wang, Tan, and Prabowo [25] showed that the covering radius of RM (3,7) in RM(4, 7) is 20. Hou [12] showed that the covering radius of RM (4,8) in RM (5,8) is at least 22.…”

Section: Introductionmentioning

confidence: 99%

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The Covering Radius of the Reed-Muller Code $RM(m-4,m)$ in $RM(m-3,m)$

Dougherty¹,

Mauldin²,

Tiefenbruck³

2021

Preprint

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We present methods for computing the distance from a Boolean polynomial on m variables of degree m − 3 (i.e., a member of the Reed-Muller code RM (m − 3, m)) to the space of lower-degree polynomials (RM (m − 4, m)). The methods give verifiable certificates for both the lower and upper bounds on this distance. By applying these methods to representative lists of polynomials, we show that the covering radius of RM (4, 8) in RM (5, 8) is 26 and the covering radius of RM (5, 9) in RM (6, 9) is between 28 and 32 inclusive, and we get improved lower bounds for higher m. We also apply our methods to various polynomials in the literature, thereby improving the known bounds on the distance from 2-resilient polynomials to RM (m − 4, m).

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“…These computations give a new (and quickly verifiable) proof of the result of Wang, Tan, and Prabowo [25] that the covering radius of RM (3,7) in RM(4, 7) is 20.…”

Section: Polynomials In 7 or Variablesmentioning

confidence: 58%

“…It is known that ρ(2, 1, 5) = 8 [13] and ρ(2, 2, 6) = 16 [1]. Reference [1] also gives the bounds 16 ≤ ρ(2, 3, 7) ≤ 22 and 16 ≤ ρ(2, 4, 8), while [25] gives ρ(2, 3, 7) ≤ 20.…”

Section: Parametersmentioning

confidence: 99%

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The Covering Radius of the Reed-Muller Code $RM(m-4,m)$ in $RM(m-3,m)$

Dougherty¹,

Mauldin²,

Tiefenbruck³

2021

Preprint

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show abstract

“…where A ∈ GL 6 (F 2 ) and g ∈ F h f un 1 (28). Therefore, N F h f un 1 (28) = 64 ≥ N F h f un i 2 (15) and i 2 = 11.…”

Section: Preliminariesmentioning

confidence: 98%

“…For n ≥ 7, the covering radius of RM (3, n) is also unknown [22]. In [28], the authors proved that the covering radius of RM (3,7) in RM (4, 7) is 20.…”

Section: Introductionmentioning

confidence: 99%

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

Wang

2019

Discrete Mathematics

Self Cite

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It was proved by J. Schatz that the covering radius of the second order Reed-Muller code RM (2, 6) is 18 (IEEE Trans Inf Theory 27: 529-530, 1985). However, the covering radius of RM (2, 7) has been an open problem for many years. In this paper, we prove that the covering radius of RM (2, 7) is 40, which is the same as the covering radius of RM (2, 7) in RM (3, 7). As a corollary, we also find new upper bounds for RM (2, n), n = 8, 9, 10.

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A trigonometric sum sharp estimate and new bounds on the nonlinearity of some cryptographic Boolean functions

Wang

Stănică

2018

Des. Codes Cryptogr.

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On the covering radius of the third order Reed–Muller code RM(3, 7)

Cited by 10 publications

References 18 publications

The Covering Radius of the Reed-Muller Code $RM(m-4,m)$ in $RM(m-3,m)$

The Covering Radius of the Reed-Muller Code $RM(m-4,m)$ in $RM(m-3,m)$

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

A trigonometric sum sharp estimate and new bounds on the nonlinearity of some cryptographic Boolean functions

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