American Institute of Mathematical Sciences

Advanced
August  2020, 14(3): 437-453. doi: 10.3934/amc.2020047

A construction of $ \mathbb{F}_2 $-linear cyclic, MDS codes

Sara D. Cardell 1, , Joan-Josep Climent 2, , Daniel Panario 3, and  Brett Stevens 3,

1. 

Instituto de Matemática, Estatística e Computação Científica, Universidade Estadual de Campinas, Campinas, Brazil
2. 

Departament de Matemàtiques, Universitat d'Alacant, Alacant, Spain
3. 

School of Mathematics and Statistics, Carleton University, Ottawa, Canada

Received  November 2018 Revised  June 2019 Published  November 2019

Fund Project: The first author was supported by CAPES (Brazil). The work of the second author was partially supported by Spanish grants AICO/2017/128 of the Generalitat Valenciana and VIGROB-287 of the Universitat d'Alacant. The third and fourth authors were supported by NSERC (Canada). The first, third and fourth authors acknowledge support from FAPESP SPRINT grant 2016/50476-0

In this paper we construct $ \mathbb{F}_2 $-linear codes over $ \mathbb{F}_{2}^{b} $ with length $ n $ and dimension $ n-r $ where $ n = rb $. These codes have good properties, namely cyclicity, low density parity-check matrices and maximum distance separation in some cases. For the construction, we consider an odd prime $ p $, let $ n = p-1 $ and utilize a partition of $ \mathbb{Z}_n $. Then we apply a Zech logarithm to the elements of these sets and use the results to construct an index array which represents the parity-check matrix of the code. These codes are always cyclic and the density of the parity-check and the generator matrices decreases to $ 0 $ as $ n $ grows (for a fixed $ r $). When $ r = 2 $ we prove that these codes are always maximum distance separable. For higher $ r $ some of them retain this property.

Keywords: $ \mathbb{F}_2 $-linear code, cyclicity, MDS codes, low density parity-check matrix, index array.
Mathematics Subject Classification: Primary: 94B05, 94B15; Secondary: 94B60.
Citation: Sara D. Cardell, Joan-Josep Climent, Daniel Panario, Brett Stevens. A construction of $ \mathbb{F}_2 $-linear cyclic, MDS codes. Advances in Mathematics of Communications, 2020, 14 (3) : 437-453. doi: 10.3934/amc.2020047
References:
[1]

M. Barbier, C. Chabot and G. Quintin, On quasi-cyclic codes as a generalization of cyclic codes, Finite Fields and Their Applications, 18 (2012), 904-919.  doi: 10.1016/j.ffa.2012.06.003.  Google Scholar

[2]

R. Barbulescu, C. Bouvier, J. Detrey, P. Gaudry, H. Jeljeli, E. Thomé, M. Videau and P. Zimmermann, Discrete logarithm in $GF(2^{809})$ with FFS, PKC 2014: Public-Key Cryptography-PKC 2014, (2014), 221–238, http://dx.doi.org/10.1007/978-3-642-54631-0_13. doi: 10.1007/978-3-642-54631-0.  Google Scholar

[3]

M. Blaum, J. Brady, J. Bruck and J. Menon, EVENODD: An efficient scheme for tolerating double disk failures in RAID architectures, IEEE Transactions on Computers, 44 (1995), 192-202.  doi: 10.1109/12.364531.  Google Scholar

[4]

M. Blaum and J. Bruck, Decoding the Golay code with Venn diagrams, IEEE Transactions on Information Theory, 36 (1990), 906-910.  doi: 10.1109/18.53756.  Google Scholar

[5]

M. Blaum, J. Bruck and A. Vardy, MDS array codes with independent parity symbols, IEEE Transactions on Information Theory, 42 (1995), 529-542.  doi: 10.1109/ISIT.1995.535761.  Google Scholar

[6]

M. Blaum, P. G. Farrell and H. C. A. van Tilborg, Array codes, in Handbook of Coding Theory, North-Holland, Amsterdam, 1/2 (1998), 1855–1909.  Google Scholar

[7]

M. Blaum and R. M. Roth, New array codes for multiple phased burst correction, IEEE Transactions on Information Theory, 39 (1993), 66-77.  doi: 10.1109/18.179343.  Google Scholar

[8]

M. Blaum and R. M. Roth, On lowest density MDS codes, IEEE Transactions on Information Theory, 45 (1999), 46-59.  doi: 10.1109/18.746771.  Google Scholar

[9]

S. D. Cardell, Constructions of MDS Codes over Extension Alphabets, PhD thesis, Departamento de Ciencia de la Computación e Inteligencia Artificial, Universidad de Alicante, Alicante, España, 2012. Google Scholar

[10]

S. D. Cardell, J.-J. Climent and V. Requena, A construction of MDS array codes, WIT Transactions on Information and Communication Technologies, 45 (2013), 47-58.  doi: 10.2495/DATA130051.  Google Scholar

[11]

S. D. Cardell and A. Fúster-Sabater, Recovering decimation-based cryptographic sequences by means of linear CAs, (2018), https://arXiv.org/abs/1802.02206. Google Scholar

[12] G. A. Gibson, Redundant Disk Arrays: Reliable, Parallel Secondary Storage, Cambridge, MA: MIT Press, 1992.   Google Scholar
[13]

K. Huber, Some comments on Zech's logarithms, IEEE Transactions on Information Theory, 36 (1990), 946-950.  doi: 10.1109/18.53764.  Google Scholar

[14]

A. Kotzig, Hamilton graphs and Hamilton circuits, Theory of Graphs and its Applications, Publ. House Czechoslovak Acad. Sci., Prague, (1964), 63–82.  Google Scholar

[15]

E. Louidor and R. M. Roth, Lowest density MDS codes over extension alphabets, IEEE Transactions on Information Theory, 52 (2006), 3186-3197.  doi: 10.1109/TIT.2006.876235.  Google Scholar

[16]

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. I, North-Holland Mathematical Library, Vol. 16. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.  Google Scholar

[17]

E. Mendelsohn and A. Rosa, One-factorizations of the complete graph-A survey, Journal of Graph Theory, 9 (1985), 43-65.  doi: 10.1002/jgt.3190090104.  Google Scholar

[18]

M. Sudan, Algorithmic Introduction to Coding Theory, 2002, https://people.csail.mit.edu/madhu/FT02/scribe/lect16.ps. Google Scholar

[19]

L. H. Xu and J. Bruck, X-code: MDS array codes with optimal encoding, IEEE Transactions on Information Theory, 45 (1999), 272-276.  doi: 10.1109/18.746809.  Google Scholar

[20]

G. V. Zaitzev, V. A. Zinov'ev and N. V. Semakov, Minimum-check-density codes for correcting bytes of errors, erasures, or defects, Problems of Information Transmission, 19 (1983), 197-204.   Google Scholar

show all references

References:
[1]

M. Barbier, C. Chabot and G. Quintin, On quasi-cyclic codes as a generalization of cyclic codes, Finite Fields and Their Applications, 18 (2012), 904-919.  doi: 10.1016/j.ffa.2012.06.003.  Google Scholar

[2]

R. Barbulescu, C. Bouvier, J. Detrey, P. Gaudry, H. Jeljeli, E. Thomé, M. Videau and P. Zimmermann, Discrete logarithm in $GF(2^{809})$ with FFS, PKC 2014: Public-Key Cryptography-PKC 2014, (2014), 221–238, http://dx.doi.org/10.1007/978-3-642-54631-0_13. doi: 10.1007/978-3-642-54631-0.  Google Scholar

[3]

M. Blaum, J. Brady, J. Bruck and J. Menon, EVENODD: An efficient scheme for tolerating double disk failures in RAID architectures, IEEE Transactions on Computers, 44 (1995), 192-202.  doi: 10.1109/12.364531.  Google Scholar

[4]

M. Blaum and J. Bruck, Decoding the Golay code with Venn diagrams, IEEE Transactions on Information Theory, 36 (1990), 906-910.  doi: 10.1109/18.53756.  Google Scholar

[5]

M. Blaum, J. Bruck and A. Vardy, MDS array codes with independent parity symbols, IEEE Transactions on Information Theory, 42 (1995), 529-542.  doi: 10.1109/ISIT.1995.535761.  Google Scholar

[6]

M. Blaum, P. G. Farrell and H. C. A. van Tilborg, Array codes, in Handbook of Coding Theory, North-Holland, Amsterdam, 1/2 (1998), 1855–1909.  Google Scholar

[7]

M. Blaum and R. M. Roth, New array codes for multiple phased burst correction, IEEE Transactions on Information Theory, 39 (1993), 66-77.  doi: 10.1109/18.179343.  Google Scholar

[8]

M. Blaum and R. M. Roth, On lowest density MDS codes, IEEE Transactions on Information Theory, 45 (1999), 46-59.  doi: 10.1109/18.746771.  Google Scholar

[9]

S. D. Cardell, Constructions of MDS Codes over Extension Alphabets, PhD thesis, Departamento de Ciencia de la Computación e Inteligencia Artificial, Universidad de Alicante, Alicante, España, 2012. Google Scholar

[10]

S. D. Cardell, J.-J. Climent and V. Requena, A construction of MDS array codes, WIT Transactions on Information and Communication Technologies, 45 (2013), 47-58.  doi: 10.2495/DATA130051.  Google Scholar

[11]

S. D. Cardell and A. Fúster-Sabater, Recovering decimation-based cryptographic sequences by means of linear CAs, (2018), https://arXiv.org/abs/1802.02206. Google Scholar

[12] G. A. Gibson, Redundant Disk Arrays: Reliable, Parallel Secondary Storage, Cambridge, MA: MIT Press, 1992.   Google Scholar
[13]

K. Huber, Some comments on Zech's logarithms, IEEE Transactions on Information Theory, 36 (1990), 946-950.  doi: 10.1109/18.53764.  Google Scholar

[14]

A. Kotzig, Hamilton graphs and Hamilton circuits, Theory of Graphs and its Applications, Publ. House Czechoslovak Acad. Sci., Prague, (1964), 63–82.  Google Scholar

[15]

E. Louidor and R. M. Roth, Lowest density MDS codes over extension alphabets, IEEE Transactions on Information Theory, 52 (2006), 3186-3197.  doi: 10.1109/TIT.2006.876235.  Google Scholar

[16]

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. I, North-Holland Mathematical Library, Vol. 16. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.  Google Scholar

[17]

E. Mendelsohn and A. Rosa, One-factorizations of the complete graph-A survey, Journal of Graph Theory, 9 (1985), 43-65.  doi: 10.1002/jgt.3190090104.  Google Scholar

[18]

M. Sudan, Algorithmic Introduction to Coding Theory, 2002, https://people.csail.mit.edu/madhu/FT02/scribe/lect16.ps. Google Scholar

[19]

L. H. Xu and J. Bruck, X-code: MDS array codes with optimal encoding, IEEE Transactions on Information Theory, 45 (1999), 272-276.  doi: 10.1109/18.746809.  Google Scholar

[20]

G. V. Zaitzev, V. A. Zinov'ev and N. V. Semakov, Minimum-check-density codes for correcting bytes of errors, erasures, or defects, Problems of Information Transmission, 19 (1983), 197-204.   Google Scholar

Table 1.  Index array constructed in Section 3
$0 $ $1 $ $2 $ $\cdots$ $p-3 $ $p-2 $
$D_0 $ $ D_0+1 $ $ D_0+2 $ $\cdots$ $ D_0+p-3 $ $D_0+p-2 $
$ D_1 $ $ D_1+1 $ $ D_1+2 $ $\cdots$ $ D_1+p-3 $ $D_1+p-2$
$\vdots$ $\vdots$ $ \vdots $ $ $ $ \vdots $ $ \vdots $
$ D_{u-1} $ $D_{u-1}+1$ $D_{u-1}+2 $ $\cdots$ $ D_{u-1}+p-3 $ $D_{u-1}+p-2 $
$D_{u+1}$ $ D_{u+1}+1 $ $D_{u+1}+2$ $\cdots$ $ D_{u+1}+p-3 $ $D_{u+1}+p-2 $
$\vdots$ $\vdots$ $ \vdots $ $ $ $ \vdots $ $ \vdots $
$ D_{b-1} $ $ D_{b-1}+1 $ $D_{b-1}+2 $ $ \cdots $ $ D_{b-1}+p-3 $ $D_{b-1}+p-2 $
Table Options

Download as excel

$0 $ $1 $ $2 $ $\cdots$ $p-3 $ $p-2 $
$D_0 $ $ D_0+1 $ $ D_0+2 $ $\cdots$ $ D_0+p-3 $ $D_0+p-2 $
$ D_1 $ $ D_1+1 $ $ D_1+2 $ $\cdots$ $ D_1+p-3 $ $D_1+p-2$
$\vdots$ $\vdots$ $ \vdots $ $ $ $ \vdots $ $ \vdots $
$ D_{u-1} $ $D_{u-1}+1$ $D_{u-1}+2 $ $\cdots$ $ D_{u-1}+p-3 $ $D_{u-1}+p-2 $
$D_{u+1}$ $ D_{u+1}+1 $ $D_{u+1}+2$ $\cdots$ $ D_{u+1}+p-3 $ $D_{u+1}+p-2 $
$\vdots$ $\vdots$ $ \vdots $ $ $ $ \vdots $ $ \vdots $
$ D_{b-1} $ $ D_{b-1}+1 $ $D_{b-1}+2 $ $ \cdots $ $ D_{b-1}+p-3 $ $D_{b-1}+p-2 $
Table 2.  Index array representing the parity-check matrix when $ r = 2 $
$0 $ $1 $ $2 $ $\cdots $ $p-3 $ $p-2 $
$ D_1 $ $ D_1+1 $ $ D_1+2 $ $ \cdots $ $ D_1+p-3 $ $D_1+p-2$
$ D_2 $ $ D_2+1 $ $ D_2+2 $ $ \cdots $ $ D_2+p-3 $ $D_2+p-2 $
$ \vdots $ $ \vdots $ $ \vdots $ $ $ $ \vdots $ $ \vdots $
$ D_{b-1} $ $ D_{b-1}+1 $ $D_{b-1}+2 $ $ \cdots $ $ D_{b-1}+p-3 $ $D_{b-1}+p-2 $
Table Options

Download as excel

$0 $ $1 $ $2 $ $\cdots $ $p-3 $ $p-2 $
$ D_1 $ $ D_1+1 $ $ D_1+2 $ $ \cdots $ $ D_1+p-3 $ $D_1+p-2$
$ D_2 $ $ D_2+1 $ $ D_2+2 $ $ \cdots $ $ D_2+p-3 $ $D_2+p-2 $
$ \vdots $ $ \vdots $ $ \vdots $ $ $ $ \vdots $ $ \vdots $
$ D_{b-1} $ $ D_{b-1}+1 $ $D_{b-1}+2 $ $ \cdots $ $ D_{b-1}+p-3 $ $D_{b-1}+p-2 $
Table 3.  Index array representing the generator matrix computed from the index array in Table 2
$B $ $ B+(b-1) $ $ B+2(b-1) $ $ B+3(b-1) $ $ B+4(b-1) $ $ \cdots $ $ B+(n-1)(b-1) $
$ 0 $ $ b-1 $ $ 2(b-1) $ $ 3(b-1) $ $ 4(b-1) $ $ \cdots $ $(n-1)(b-1) $
$ 1 $ $ b $ $ 2(b-1)+1 $ $ 3(b-1)+1 $ $ 4(b-1)+1 $ $ \cdots $ $(n-1)(b-1)+1 $
$ 2 $ $ b+1 $ $ 2(b-1)+2 $ $ 3(b-1)+2 $ $ 4(b-1)+2 $ $ \cdots $ $(n-1)(b-1)+2 $
$ \vdots $ $ \vdots $ $ \vdots $ $ \cdots $ $ \vdots $ $ $ $\vdots $
$ b-2 $ $ 2b-3 $ $ 3(b-1)-1 $ $ 4(b-1)-1 $ $ 5(b-1)-1 $ $ \ldots $ $n(b-1)-1 $
Table Options

Download as excel

$B $ $ B+(b-1) $ $ B+2(b-1) $ $ B+3(b-1) $ $ B+4(b-1) $ $ \cdots $ $ B+(n-1)(b-1) $
$ 0 $ $ b-1 $ $ 2(b-1) $ $ 3(b-1) $ $ 4(b-1) $ $ \cdots $ $(n-1)(b-1) $
$ 1 $ $ b $ $ 2(b-1)+1 $ $ 3(b-1)+1 $ $ 4(b-1)+1 $ $ \cdots $ $(n-1)(b-1)+1 $
$ 2 $ $ b+1 $ $ 2(b-1)+2 $ $ 3(b-1)+2 $ $ 4(b-1)+2 $ $ \cdots $ $(n-1)(b-1)+2 $
$ \vdots $ $ \vdots $ $ \vdots $ $ \cdots $ $ \vdots $ $ $ $\vdots $
$ b-2 $ $ 2b-3 $ $ 3(b-1)-1 $ $ 4(b-1)-1 $ $ 5(b-1)-1 $ $ \ldots $ $n(b-1)-1 $
Table 4.  Index array obtained by subtracting $ 1 $ modulo $ n $ to every set in the array given in Table 1
$p-2 $ $0 $ $1 $ $2 $ $\cdots $ $p-3 $
$ D_0+p-2 $ $D_0 $ $D_0+1 $ $ D_0+2 % $ $ \cdots $ $ D_0+p-3 $
$ D_1+p-2 $ $D_1 $ $D_1 +1 $ $ D_1+2 % $ $ \cdots $ $ D_1+p-3$
$ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ $ $ \vdots $
$ D_{u-1}+p-2 $ $D_{u-1} $ $D_{u-1}+1 $ $ D_{u-1}+2 % $ $ \cdots $ $ D_{u-1}+p-3 $
$ D_{u+1}+p-2 $ $D_{u+1} $ $D_{u+1} +1 $ $ D_{u+1}+2 % $ $ \cdots $ $ D_{u+1}+p-3 $
$ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ $ $ \vdots $
$ D_{b-1}+p-2 $ $D_{b-1} $ $D_{b-1} +1 $ $ D_{b-1}+2 $ $ \cdots $ $ D_{b-1}+p-3 $
Table Options

Download as excel

$p-2 $ $0 $ $1 $ $2 $ $\cdots $ $p-3 $
$ D_0+p-2 $ $D_0 $ $D_0+1 $ $ D_0+2 % $ $ \cdots $ $ D_0+p-3 $
$ D_1+p-2 $ $D_1 $ $D_1 +1 $ $ D_1+2 % $ $ \cdots $ $ D_1+p-3$
$ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ $ $ \vdots $
$ D_{u-1}+p-2 $ $D_{u-1} $ $D_{u-1}+1 $ $ D_{u-1}+2 % $ $ \cdots $ $ D_{u-1}+p-3 $
$ D_{u+1}+p-2 $ $D_{u+1} $ $D_{u+1} +1 $ $ D_{u+1}+2 % $ $ \cdots $ $ D_{u+1}+p-3 $
$ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ $ $ \vdots $
$ D_{b-1}+p-2 $ $D_{b-1} $ $D_{b-1} +1 $ $ D_{b-1}+2 $ $ \cdots $ $ D_{b-1}+p-3 $
Table 5.  MDS property of $ C(p, r, \alpha) $ for different values of $ p $ and $ r $
$p $
5 7 11 13 17 19 23 29 31 37 41 43
$ r $ 2 √ √ √ √ √ √ √ √ √ √ √ √
3 × √ √ √ √ √
4 × × √ √ ×
5 × × ×
6 × × × × ×
7 × ×
8 × ×
9 × ×
10 × ×
11 ×
12 ×
13
14 × ×
15 ×
Table Options

Download as excel

$p $
5 7 11 13 17 19 23 29 31 37 41 43
$ r $ 2 √ √ √ √ √ √ √ √ √ √ √ √
3 × √ √ √ √ √
4 × × √ √ ×
5 × × ×
6 × × × × ×
7 × ×
8 × ×
9 × ×
10 × ×
11 ×
12 ×
13
14 × ×
15 ×
[1]

Fengwei Li, Qin Yue, Xiaoming Sun. The values of two classes of Gaussian periods in index 2 case and weight distributions of linear codes. Advances in Mathematics of Communications, 2021, 15 (1) : 131-153. doi: 10.3934/amc.2020049
[2]

Hongwei Liu, Jingge Liu. On $ \sigma $-self-orthogonal constacyclic codes over $ \mathbb F_{p^m}+u\mathbb F_{p^m} $. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020127
[3]

Hai Q. Dinh, Bac T. Nguyen, Paravee Maneejuk. Constacyclic codes of length $ 8p^s $ over $ \mathbb F_{p^m} + u\mathbb F_{p^m} $. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020123
[4]

Yuan Cao, Yonglin Cao, Hai Q. Dinh, Ramakrishna Bandi, Fang-Wei Fu. An explicit representation and enumeration for negacyclic codes of length $ 2^kn $ over $ \mathbb{Z}_4+u\mathbb{Z}_4 $. Advances in Mathematics of Communications, 2021, 15 (2) : 291-309. doi: 10.3934/amc.2020067
[5]

Parikshit Upadhyaya, Elias Jarlebring, Emanuel H. Rubensson. A density matrix approach to the convergence of the self-consistent field iteration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 99-115. doi: 10.3934/naco.2020018
[6]

Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, 2021, 15 (1) : 159-183. doi: 10.3934/ipi.2020076
[7]

Vito Napolitano, Ferdinando Zullo. Codes with few weights arising from linear sets. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020129
[8]

Dandan Wang, Xiwang Cao, Gaojun Luo. A class of linear codes and their complete weight enumerators. Advances in Mathematics of Communications, 2021, 15 (1) : 73-97. doi: 10.3934/amc.2020044
[9]

Huyuan Chen, Dong Ye, Feng Zhou. On gaussian curvature equation in $ \mathbb{R}^2 $ with prescribed nonpositive curvature. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3201-3214. doi: 10.3934/dcds.2020125
[10]

Guoyuan Chen, Yong Liu, Juncheng Wei. Nondegeneracy of harmonic maps from $ {{\mathbb{R}}^{2}} $ to $ {{\mathbb{S}}^{2}} $. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3215-3233. doi: 10.3934/dcds.2019228
[11]

Shudi Yang, Xiangli Kong, Xueying Shi. Complete weight enumerators of a class of linear codes over finite fields. Advances in Mathematics of Communications, 2021, 15 (1) : 99-112. doi: 10.3934/amc.2020045
[12]

Jong Yoon Hyun, Boran Kim, Minwon Na. Construction of minimal linear codes from multi-variable functions. Advances in Mathematics of Communications, 2021, 15 (2) : 227-240. doi: 10.3934/amc.2020055
[13]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447
[14]

Yueh-Cheng Kuo, Huan-Chang Cheng, Jhih-You Syu, Shih-Feng Shieh. On the nearest stable $ 2\times 2 $ matrix, dedicated to Prof. Sze-Bi Hsu in appreciation of his inspiring ideas. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020358
[15]

Ivan Bailera, Joaquim Borges, Josep Rifà. On Hadamard full propelinear codes with associated group $ C_{2t}\times C_2 $. Advances in Mathematics of Communications, 2021, 15 (1) : 35-54. doi: 10.3934/amc.2020041
[16]

Hongming Ru, Chunming Tang, Yanfeng Qi, Yuxiao Deng. A construction of $ p $-ary linear codes with two or three weights. Advances in Mathematics of Communications, 2021, 15 (1) : 9-22. doi: 10.3934/amc.2020039
[17]

Lei Liu, Li Wu. Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020378
[18]

Peizhao Yu, Guoshan Zhang, Yi Zhang. Decoupling of cubic polynomial matrix systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 13-26. doi: 10.3934/naco.2020012
[19]

Shengxin Zhu, Tongxiang Gu, Xingping Liu. AIMS: Average information matrix splitting. Mathematical Foundations of Computing, 2020, 3 (4) : 301-308. doi: 10.3934/mfc.2020012
[20]

Mingchao Zhao, You-Wei Wen, Michael Ng, Hongwei Li. A nonlocal low rank model for poisson noise removal. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021003

2019 Impact Factor: 0.734

Tools

Metrics

  • PDF downloads (2512)
  • HTML views (438)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences