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The Covering Radius of the Reed--Muller Code RM(2,7) is 40 [article]
2018 arXiv pre-print
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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). ... 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. ... Acknowledgment The first author would like to thank the financial support from the National Natural Science Foundation of China (Grant 61572189). ...
arXiv:1809.04864v1 fatcat:xmye5myc4bgq7dl4nzejmzmd2a
The Covering Radius of the Third-Order Reed-Muller Code RM(3,7) is 20 [article]
2023 arXiv pre-print
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We prove the covering radius of the third-order Reed-Muller code RM(3,7) is 20, which was previously known to be between 20 and 23 (inclusive). ... The covering radius of RM(3, 7) is the maximum third-order nonlinearity among all 7-variable Boolean functions. ... Acknowledgements We thank the editor and anonymous reviewers for their valuable comments, for example, for their suggestions to present the results in more general forms. ...
arXiv:2206.10881v3 fatcat:qoiensle6zdofjpza5sh7ou7nu
On the Covering Radius of Second Order Binary Reed-Muller Code in the Set of Resilient Boolean Functions [chapter]
2003 Lecture Notes in Computer Science
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Second, we show that the covering radius of the binary Reed-Muller code RM (2, 7) in the set R3,7 is 32. ... Let Rt,n denote the set of t-resilient Boolean functions of n variables. First, we prove that the covering radius of the binary Reed-Muller code RM (2, 6) in the sets Rt,6, t = 0, 1, 2 is 16. ... In Sect. 3 we prove that the covering radius of the binary Reed-Muller code RM (2, 6) in the sets R t,6 , t = 0, 1, 2 is 16 and in Sect. 4 we present a proof that the covering radius of the binary Reed-Muller ...
doi:10.1007/978-3-540-40974-8_8 fatcat:2cvy3j4m4rf6hgjfovbx353niy
On the Covering Radius of the Second Order Reed-Muller Code of Length 128 [article]
2015 arXiv pre-print
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In 1981, Schatz proved that the covering radius of the binary Reed-Muller code RM(2,6) is 18. For RM(2,7), we only know that its covering radius is between 40 and 44. ... In this paper, we prove that the covering radius of the binary Reed-Muller code RM(2,7) is at most 42. ... Introduction In [11] , Schatz proved that the covering radius of the binary Reed-Muller code RM (2, 6) is 18. For m ≥ 7, the covering radius of RM (2, m) is still unknown. ...
arXiv:1510.08535v1 fatcat:wlajkrbiw5gyrdeaaoeg26gd7u
Classification of some cosets of Reed-Muller codes [article]
2022 arXiv pre-print
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This note presents a descending method that allows us to classify quotients of Reed-Muller codes of lenghth 128 under the action of the affine general linear group. ... 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) . ... 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] . ...
arXiv:2208.02469v2 fatcat:uvdc5kgvgbchzi7boirkkr77ae
On metric regularity of Reed-Muller codes [article]
2020 arXiv pre-print
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Combined with previous results by Tokareva N. (2012) concerning duality of affine and bent functions, this establishes the metric regularity of most Reed-Muller codes with known covering radius. ... It is conjectured that all Reed-Muller codes are metrically regular. ... covering radius of the code RM (2, 7) to be equal to 40 [19] . ...
arXiv:1912.10811v2 fatcat:rvw5etqerbgwhohqhfyd5eeala
New Covering Radius of Reed–Muller Codes for$t$-Resilient Functions
2004 IEEE Transactions on Information Theory
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From this point of view, we define a new covering radiusρ(t, r, n) as the maximum distance between a t-resilient function f (X) and the r-th order Reed-Muller code RM (r, n). ... Finally, we present a table of numerical bounds forρ(t, r, n). ... The r-th order Reed-Muller code RM (r, n) is identical to the set of n-variable Boolean function g(X) such that deg(g) ≤ r. ...
doi:10.1109/tit.2004.824913 fatcat:al54dy7egvfwfez24ognvj3evq
Page 3146 of Mathematical Reviews Vol. , Issue 2003d [page]
2003 Mathematical Reviews
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Despite much research over the last quarter-century, the covering radius of RM(1, 7) is largely unknown when mm is odd: only lower bounds are known for m > 7. ... The author focuses on “almost opti- mal” cosets, those with minimum weight at least 2”~! — 2¢"-))/2, a known lower bound on the covering radius of RM(1, 7). ...
Page 7633 of Mathematical Reviews Vol. , Issue 2001J [page]
2001 Mathematical Reviews
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2m-1 4 2’"—-0/2 2") and RM(1,m) C@ CRM(2,m), where RM(r,m) denotes a binary Reed-Muller code. In this paper it is shown that for m = 5 all optimal codes are Gold codes. ... If this weight is k, they ask for the least integer r such that every vector of weight & is within distance r of some codeword. This r is their covering radius of the code. ...
Perfect mixed codes from generalized Reed-Muller codes [article]
2023 arXiv pre-print
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The construction we propose is based on partitions of distance-2 MDS codes into Reed-Muller-like codes of order (q - 1)m - 2. ... Codes whose parameters are the same as the parameters of generalized Reed-Muller codes are called Reed-Muller-like codes. ... The covering radius ρ(C) of a code C of length n is the minimum number ρ ∈ {0, 1, . . . , n} such that ∪ c∈C B ρ (c) = V n . ...
arXiv:2312.15937v1 fatcat:ooj76c7hlzey3af3r7jeesnqpe
Page 1460 of Mathematical Reviews Vol. , Issue 99b [page]
1991 Mathematical Reviews
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Math. 37 (1979), no. 2, 419-422; MR 81a:94034] of the covering radius of RM(m — 3,m). ... Theory 40 (1994), no. 5, 1406-1416; MR 95h:94034] proved that there is no binary [33,24] code with R=2. This result imples that 7[33,24] > 2. ...
Covering radius---Survey and recent results
1985 IEEE Transactions on Information Theory
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There are a number of upper and lower bounds, including asymptotic results, a few exact determinations of covering radius, some extensive relations with other aspects of coding theory through the Reed-Muller ... There is also a recent result on the complexity of computing the covering radius. ... 11) For r + m -1, does the Reed-Muller code RM(r, m) have even covering radius? ...
doi:10.1109/tit.1985.1057043 fatcat:bq2zv7acnbapta45kd7lkbca7u
Page 3115 of Mathematical Reviews Vol. , Issue 95e [page]
1995 Mathematical Reviews
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cosets of the Reed-Muller code R(m — 3, m). ... The covering radius is defined as the weight of the coset of greatest weight; it is the maximum distance of any vector in the space from the code. In a paper on RM, G. Seroussi and A. ...
Weighted Reed-Muller codes revisited [article]
2011 arXiv pre-print
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In conclusion the weighted Reed-Muller code construction is much better than its reputation. ... For a class of affine variety codes that contains the weighted Reed-Muller codes we then present two list decoding algorithms. ... Weighted Reed-Muller codes The first example of codes E(M, S) that comes to mind are the q-ary Reed-Muller codes RM q (u, m). ...
arXiv:1108.6185v1 fatcat:fw5ylmg7wnbtxmnr3wmuwmgffe
Weighted Reed–Muller codes revisited
2012 Designs, Codes and Cryptography
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In conclusion the weighted Reed-Muller code construction is much better than its reputation. ... For a class of affine variety codes that contains the weighted Reed-Muller codes we then present two list decoding algorithms. ... Weighted Reed-Muller codes The first example of codes E(M, S) that comes to mind are the q-ary Reed-Muller codes RM q (u, m). ...
doi:10.1007/s10623-012-9680-8 fatcat:atvj22ye7rgr3jgxwqiaqbz6xm
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