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Qichun Wang - One of the best experts on this subject based on the ideXlab platform.
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the Covering Radius of the reed muller code rm 2 7 is 40
Discrete Mathematics, 2019Co-Authors: Qichun WangAbstract:Abstract It was proved by J. Schatz that the Covering Radius of the second order Reed–Muller code R M ( 2 , 6 ) is 18 (Schatz (1981)). However, the Covering Radius of R M ( 2 , 7 ) has been an open problem for many years. In this paper, we prove that the Covering Radius of R M ( 2 , 7 ) is 40, which is the same as the Covering Radius of R M ( 2 , 7 ) in R M ( 3 , 7 ) . As a corollary, we also find new upper bounds for the Covering Radius of R M ( 2 , n ) , n = 8 , 9 , 10 .
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The Covering Radius of the Reed–Muller code RM(2,7) is 40
Discrete Mathematics, 2019Co-Authors: Qichun WangAbstract:Abstract It was proved by J. Schatz that the Covering Radius of the second order Reed–Muller code R M ( 2 , 6 ) is 18 (Schatz (1981)). However, the Covering Radius of R M ( 2 , 7 ) has been an open problem for many years. In this paper, we prove that the Covering Radius of R M ( 2 , 7 ) is 40, which is the same as the Covering Radius of R M ( 2 , 7 ) in R M ( 3 , 7 ) . As a corollary, we also find new upper bounds for the Covering Radius of R M ( 2 , n ) , n = 8 , 9 , 10 .
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The Covering Radius of the Reed--Muller Code $RM(2,7)$ is 40
arXiv: Information Theory, 2018Co-Authors: Qichun WangAbstract: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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the Covering Radius of the reed muller code rm 2 7 is 40
arXiv: Information Theory, 2018Co-Authors: Qichun WangAbstract: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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New bounds on the Covering Radius of the second order Reed-Muller code of length 128
Cryptography and Communications, 2018Co-Authors: Qichun Wang, Pantelimon StănicăAbstract:In 1981, Schatz proved that the Covering Radius of the binary Reed-Muller code RM(2, 6) is 18. It was previously shown that the Covering Radius of RM(2, 7) is between 40 and 44. In this paper, we prove that the Covering Radius of RM(2, 7) is at most 42. As a corollary, we also find new upper bounds for RM(2, n), n = 8, 9, 10. Moreover, we give a sufficient and necessary condition for the Covering Radius of RM(2, 7) to be equal to 42. Using this condition, we prove that the Covering Radius of RM(2, 7) in RM(4, 7) is exactly 40, and as a by-product, we conclude that the Covering Radius of RM(2, 7) in the set of 2-resilient Boolean functions is at most 40, which improves the bound given by Borissov et al. (IEEE Trans. Inf. Theory 51(3):1182–1189, 2005).
C. Durairajan - One of the best experts on this subject based on the ideXlab platform.
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On the Covering Radius of Codes over Z p k
Mathematics, 2020Co-Authors: Mohan Cruz, C. Durairajan, Patrick SoléAbstract:In this correspondence, we investigate the Covering Radius of various types of repetition codes over Z p k ( k ≥ 2 ) with respect to the Lee distance. We determine the exact Covering Radius of the various repetition codes, which have been constructed using the zero divisors and units in Z p k . We also derive the lower and upper bounds on the Covering Radius of block repetition codes over Z p k .
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On codes over ℤp2 and its Covering Radius
Asian-european Journal of Mathematics, 2019Co-Authors: N. Annamalai, C. DurairajanAbstract:This paper gives lower and upper bounds on the Covering Radius of codes over ℤp2 with respect to Lee distance. We also determine the Covering Radius of various repetition codes over ℤp2.
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On codes over ℤp2 and its Covering Radius
Asian-European Journal of Mathematics, 2019Co-Authors: N. Annamalai, C. DurairajanAbstract:This paper gives lower and upper bounds on the Covering Radius of codes over [Formula: see text] with respect to Lee distance. We also determine the Covering Radius of various repetition codes over [Formula: see text]
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On Covering Radius of codes over ℤ2p
Asian-european Journal of Mathematics, 2018Co-Authors: N. Annamalai, C. DurairajanAbstract:In this paper, we gives lower and upper bounds on the Covering Radius of codes over ℤ2p, where p is a prime integer with respect to Lee distance. We also determine the Covering Radius of various Re...
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On Covering Radius of codes over ℤ2p
Asian-European Journal of Mathematics, 2018Co-Authors: N. Annamalai, C. DurairajanAbstract:In this paper, we gives lower and upper bounds on the Covering Radius of codes over [Formula: see text], where [Formula: see text] is a prime integer with respect to Lee distance. We also determine the Covering Radius of various Repetition codes over [Formula: see text], where [Formula: see text] is a prime integer.
Elodie Leducq - One of the best experts on this subject based on the ideXlab platform.
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on the Covering Radius of first order generalized reed muller codes
IEEE Transactions on Information Theory, 2013Co-Authors: Elodie LeducqAbstract:We generalize to any finite Fq fields a theorem about Covering Radius of codes of strength 2 proved by Helleseth and coworkers. Then,using this result and partial Covering Radius, we give bounds for the Covering Radius of first-order generalized Reed-Muller codes. Finally, using Magma, we get some improvements for F3.
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On the Covering Radius of First-Order Generalized Reed–Muller Codes
IEEE Transactions on Information Theory, 2013Co-Authors: Elodie LeducqAbstract:We generalize to any finite Fq fields a theorem about Covering Radius of codes of strength 2 proved by Helleseth and coworkers. Then,using this result and partial Covering Radius, we give bounds for the Covering Radius of first-order generalized Reed-Muller codes. Finally, using Magma, we get some improvements for F3.
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On the Covering Radius of first order generalized Reed-Muller codes
arXiv: Number Theory, 2011Co-Authors: Elodie LeducqAbstract:We generalize to any q a theorem about Covering Radius of linear codes proved by Helleseth, Klove and Mykkelvit. Then we determine the Covering Radius of first order generalized Reed-Muller codes in second order generalized Reed-Muller codes. Using these results, we are able to give bounds for the Covering Radius of first order generalized Reed-Muller codes. Finaly, using Magma, we get some improvements for q=3.
Theo Fanuela Prabowo - One of the best experts on this subject based on the ideXlab platform.
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On the Covering Radius of the third order Reed–Muller code RM(3, 7)
Designs Codes and Cryptography, 2018Co-Authors: Qichun Wang, Chik How Tan, Theo Fanuela PrabowoAbstract:The Covering Radius of the third order Reed–Muller code of length 128 has been an open problem for many years. The best upper bound of it is known to be 22. In this paper, we give a sufficient and necessary condition for the Covering Radius of RM (3, 7) to be equal to 22. Using this condition, we prove that the Covering Radius of RM (3, 7) in RM (4, 7) is 20. Therefore, if the third-order nonlinearity of a 7-variable Boolean function is greater than 20, then its algebraic degree is at least 5. As a corollary, we conclude that the Covering Radius of RM (3, 7) in the set of 2-resilient Boolean functions is at most 20 which improves the bound given by Borissov et al. (IEEE Trans Inf Theory 51:1182–1189, 2005 ).
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on the Covering Radius of the third order reed muller code rm 3 7
Designs Codes and Cryptography, 2018Co-Authors: Qichun Wang, Chik How Tan, Theo Fanuela PrabowoAbstract:The Covering Radius of the third order Reed–Muller code of length 128 has been an open problem for many years. The best upper bound of it is known to be 22. In this paper, we give a sufficient and necessary condition for the Covering Radius of RM(3, 7) to be equal to 22. Using this condition, we prove that the Covering Radius of RM(3, 7) in RM(4, 7) is 20. Therefore, if the third-order nonlinearity of a 7-variable Boolean function is greater than 20, then its algebraic degree is at least 5. As a corollary, we conclude that the Covering Radius of RM(3, 7) in the set of 2-resilient Boolean functions is at most 20 which improves the bound given by Borissov et al. (IEEE Trans Inf Theory 51:1182–1189, 2005).
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On the Covering Radius of the third order Reed–Muller code RM(3, 7)
Designs Codes and Cryptography, 2017Co-Authors: Qichun Wang, Chik How Tan, Theo Fanuela PrabowoAbstract:The Covering Radius of the third order Reed–Muller code of length 128 has been an open problem for many years. The best upper bound of it is known to be 22. In this paper, we give a sufficient and necessary condition for the Covering Radius of RM(3, 7) to be equal to 22. Using this condition, we prove that the Covering Radius of RM(3, 7) in RM(4, 7) is 20. Therefore, if the third-order nonlinearity of a 7-variable Boolean function is greater than 20, then its algebraic degree is at least 5. As a corollary, we conclude that the Covering Radius of RM(3, 7) in the set of 2-resilient Boolean functions is at most 20 which improves the bound given by Borissov et al. (IEEE Trans Inf Theory 51:1182–1189, 2005).
Patric R J Ostergard - One of the best experts on this subject based on the ideXlab platform.
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Linear codes with Covering Radius 3
Designs Codes and Cryptography, 2009Co-Authors: Alexander A. Davydov, Patric R J OstergardAbstract:The shortest possible length of a q-ary linear code of Covering Radius R and codimension r is called the length function and is denoted by ? q (r, R). Constructions of codes with Covering Radius 3 are here developed, which improve best known upper bounds on ? q (r, 3). General constructions are given and upper bounds on ? q (r, 3) for q = 3, 4, 5, 7 and r ? 24 are tabulated.
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linear codes with Covering Radius and codimension
2001Co-Authors: Gerard D Cohen, Alexander A. Davydov, Iiro Honkala, F J Macwilliams, Elwyn R Berlekamp, Annelise Lobstein, Neil J. A. Sloane, Simon Litsyn, Patric R J OstergardAbstract:Let [ ] denote a linear code over with length , codimension , and Covering Radius . We use a modification of constructions of [2 +1 2 3] 2 and [3 +1 3 5] 3 codes ( 5) to produce infinite families of good codes with Covering Radius 2 and 3 and codimension .
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Linear codes with Covering Radius R=2, 3 and codimension tR
IEEE Transactions on Information Theory, 2001Co-Authors: Alexander A. Davydov, Patric R J OstergardAbstract:Let [n,n-r]/sub q/R denote a linear code over F/sub q/ with length n, codimension r, and Covering Radius R. We use a modification of constructions of [2q+1, 2q-3]/sub q/2 and [3q+1, 3q-5]/sub q/3 codes (q/spl ges/5) to produce infinite families of good codes with Covering Radius 2 and 3 and codimension tR.
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New Quaternary Linear Codes with Covering Radius 2
Finite Fields and Their Applications, 2000Co-Authors: Alexander A. Davydov, Patric R J OstergardAbstract:A new quaternary linear code of length 19, codimension 5, and Covering Radius 2 is found in a computer search using tabu search, a local search heuristic. Starting from this code, which has some useful partitioning properties, different lengthening constructions are applied to get an infinite family of new, record-breaking quaternary codes of Covering Radius 2 and odd codimension. An algebraic construction of Covering codes over alphabets of even characteristic is also given.
