doi: 10.3934/amc.2022042
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Optimal binary linear codes from posets of the disjoint union of two chains

1. 

School of Computer Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, China

2. 

State Key Laboratory of Integrated Services Networks, Xidian University, Xi'an 710071, China

3. 

Konkuk University, Glocal Campus, 268 Chungwon-daero Chungju-si Chungcheongbuk-do 27478, Republic of Korea

4. 

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211100, China

*Corresponding author: Yansheng Wu

Received  February 2022 Revised  April 2022 Early access June 2022

Fund Project: This work was supported by the National Natural Science Foundation of China (Nos. 12101326, 62172219), the Natural Science Foundation of Jiangsu Province (No. BK20210575), the Natural Science Research Project of Colleges and Universities in Jiangsu Province (No. 21KJB110005), the National Research Foundation of Korea(NRF) grant funded by the Korea government(MEST)(NRF-2017R1D1A1B05030707), and the Foundation of State Key Laboratory of Integrated Services Networks under Grant ISN23-22

Recently, some infinite families of optimal binary linear codes are constructed from simplicial complexes. Afterwards, the construction method was extended to using arbitrary posets. In this paper, based on a generic construction of linear codes, we obtain four classes of optimal binary linear codes by using the posets of two chains. Two of them induce Griesmer codes which are not equivalent to the linear codes constructed by Belov. Those codes are exploited to construct secret sharing schemes in cryptography as well.

Citation: Yansheng Wu, Jong Yoon Hyun, Qin Yue. Optimal binary linear codes from posets of the disjoint union of two chains. Advances in Mathematics of Communications, doi: 10.3934/amc.2022042
References:
[1]

A. R. AndersonC. DingT. Helleseth and T. Kløve, How to build robust shared control systems, Des. Codes Cryptogr., 15 (1998), 111-124.  doi: 10.1023/A:1026421315292.

[2]

A. Ashikhmin and A. Barg, Minimal vectors in linear codes, IEEE Trans. Inf. Theory, 44 (1998), 2010-2017.  doi: 10.1109/18.705584.

[3]

D. Bartoli and M. Bonini, Minimal Linear codes in odd characteristic, IEEE Trans. Inf. Theory, 65 (2019), 4152-4155.  doi: 10.1109/TIT.2019.2891992.

[4]

B. I. Belov, A conjecture on the griesmer bound, optimimization methods and their applications, Optimization Methods and Their Applications, Sibirsk. Énerget. Inst. Sibirsk. Otdel. Akad. Nauk SSSR, Irkutsk, 182 (1974), 100-106. 

[5]

B. I. BelovV. N. Logačhev and V. P. Sandimirov, Construction of a class of linear binary codes achieving the Varšhamov-Griesmer bound, Problemy Peredači Informacii, 10 (1974), 36-44. 

[6]

C. CarletC. Ding and J. Yuan, Linear codes from perfect nonlinear mappings and their secret sharing schemes, IEEE Trans. Inf. Theory, 51 (2005), 2089-2102.  doi: 10.1109/TIT.2005.847722.

[7]

C. Ding, Linear codes from some 2-designs, IEEE Trans. Inf. Theory, 61 (2015), 3265-3275.  doi: 10.1109/TIT.2015.2420118.

[8]

C. DingZ. Heng and Z. Zhou, Minimal binary linear codes, IEEE Trans. Inf. Theory, 64 (2018), 6536-6545.  doi: 10.1109/TIT.2018.2819196.

[9]

M. Grassl, Bounds on the Minimum Distance of Linear Codes, 2021, Available from: http://www.codetables.de.

[10]

J. H. Griesmer, A bound for error correcting codes, IBM J. Res. Dev., 4 (1960), 532-542.  doi: 10.1147/rd.45.0532.

[11] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511807077.
[12]

J. Y. HyunH. K. KimY. Wu and Q. Yue, Optimal minimal linear codes from posets, Des. Codes Cryptogr., 88 (2020), 2475-2492.  doi: 10.1007/s10623-020-00793-0.

[13]

Z. Heng and Q. Yue, A class of binary linear codes with at most three weights, IEEE Commun. Lett., 19 (2015), 1488-1491.  doi: 10.1109/LCOMM.2015.2455032.

[14]

Z. Heng and Q. Yue, Two classes of two-weight linear codes, Finite Fields Appl., 38 (2016), 72-92.  doi: 10.1016/j.ffa.2015.12.002.

[15]

Z. Heng and Q. Yue, Evaluation of the Hamming weights of a class of linear codes based on Gauss sums, Des. Codes Cryptogr., 83 (2017), 307-326.  doi: 10.1007/s10623-016-0222-7.

[16]

C. LiQ. Yue and F. Li, Weight distributions of cyclic codes with respect to pairwise coprime order elements, Finite Fields Appl., 28 (2014), 94-114.  doi: 10.1016/j.ffa.2014.01.009.

[17]

Y. Liu and Z. Liu, On some classes of codes with a few weights, Adv. Math. Commun., 12 (2018), 415-428.  doi: 10.3934/amc.2018025.

[18]

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

[19]

J. Neggers and H. S. Kim, Basic Posets, World Scientific Publishing Co., Inc., River Edge, NJ, 1998. doi: 10.1142/3890.

[20]

C. TangY. QiuQ. Liao and Z. Zhou, Full characterization of minimal linear codes as cutting blocking sets, IEEE Trans. Inf. Theory, 67 (2021), 3690-3700.  doi: 10.1109/TIT.2021.3070377.

[21]

S. YangX. Kong and C. Tang, A construction of linear codes and their complete weight enumerators, Finite Fields Appl., 48 (2017), 196-226.  doi: 10.1016/j.ffa.2017.08.001.

[22]

J. Yuan and C. Ding, Secret sharing schemes from three classes of linear codes, IEEE Trans. Inf. Theory, 52 (2006), 206-212.  doi: 10.1109/TIT.2005.860412.

[23]

Z. ZhouC. TangX. Li and C. Ding, Binary LCD codes and self-orthogonal codes from a generic construction, IEEE Trans. Inf. Theory, 65 (2019), 16-27.  doi: 10.1109/TIT.2018.2823704.

[24]

W. ZhangH. Yan and H. Wei, Four families of minimal binary linear codes with $w_{min}/w_{max}\leq 1/2$, Appl. Algebra Engrg. Comm. Comput., 30 (2019), 175-184.  doi: 10.1007/s00200-018-0367-x.

show all references

References:
[1]

A. R. AndersonC. DingT. Helleseth and T. Kløve, How to build robust shared control systems, Des. Codes Cryptogr., 15 (1998), 111-124.  doi: 10.1023/A:1026421315292.

[2]

A. Ashikhmin and A. Barg, Minimal vectors in linear codes, IEEE Trans. Inf. Theory, 44 (1998), 2010-2017.  doi: 10.1109/18.705584.

[3]

D. Bartoli and M. Bonini, Minimal Linear codes in odd characteristic, IEEE Trans. Inf. Theory, 65 (2019), 4152-4155.  doi: 10.1109/TIT.2019.2891992.

[4]

B. I. Belov, A conjecture on the griesmer bound, optimimization methods and their applications, Optimization Methods and Their Applications, Sibirsk. Énerget. Inst. Sibirsk. Otdel. Akad. Nauk SSSR, Irkutsk, 182 (1974), 100-106. 

[5]

B. I. BelovV. N. Logačhev and V. P. Sandimirov, Construction of a class of linear binary codes achieving the Varšhamov-Griesmer bound, Problemy Peredači Informacii, 10 (1974), 36-44. 

[6]

C. CarletC. Ding and J. Yuan, Linear codes from perfect nonlinear mappings and their secret sharing schemes, IEEE Trans. Inf. Theory, 51 (2005), 2089-2102.  doi: 10.1109/TIT.2005.847722.

[7]

C. Ding, Linear codes from some 2-designs, IEEE Trans. Inf. Theory, 61 (2015), 3265-3275.  doi: 10.1109/TIT.2015.2420118.

[8]

C. DingZ. Heng and Z. Zhou, Minimal binary linear codes, IEEE Trans. Inf. Theory, 64 (2018), 6536-6545.  doi: 10.1109/TIT.2018.2819196.

[9]

M. Grassl, Bounds on the Minimum Distance of Linear Codes, 2021, Available from: http://www.codetables.de.

[10]

J. H. Griesmer, A bound for error correcting codes, IBM J. Res. Dev., 4 (1960), 532-542.  doi: 10.1147/rd.45.0532.

[11] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511807077.
[12]

J. Y. HyunH. K. KimY. Wu and Q. Yue, Optimal minimal linear codes from posets, Des. Codes Cryptogr., 88 (2020), 2475-2492.  doi: 10.1007/s10623-020-00793-0.

[13]

Z. Heng and Q. Yue, A class of binary linear codes with at most three weights, IEEE Commun. Lett., 19 (2015), 1488-1491.  doi: 10.1109/LCOMM.2015.2455032.

[14]

Z. Heng and Q. Yue, Two classes of two-weight linear codes, Finite Fields Appl., 38 (2016), 72-92.  doi: 10.1016/j.ffa.2015.12.002.

[15]

Z. Heng and Q. Yue, Evaluation of the Hamming weights of a class of linear codes based on Gauss sums, Des. Codes Cryptogr., 83 (2017), 307-326.  doi: 10.1007/s10623-016-0222-7.

[16]

C. LiQ. Yue and F. Li, Weight distributions of cyclic codes with respect to pairwise coprime order elements, Finite Fields Appl., 28 (2014), 94-114.  doi: 10.1016/j.ffa.2014.01.009.

[17]

Y. Liu and Z. Liu, On some classes of codes with a few weights, Adv. Math. Commun., 12 (2018), 415-428.  doi: 10.3934/amc.2018025.

[18]

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

[19]

J. Neggers and H. S. Kim, Basic Posets, World Scientific Publishing Co., Inc., River Edge, NJ, 1998. doi: 10.1142/3890.

[20]

C. TangY. QiuQ. Liao and Z. Zhou, Full characterization of minimal linear codes as cutting blocking sets, IEEE Trans. Inf. Theory, 67 (2021), 3690-3700.  doi: 10.1109/TIT.2021.3070377.

[21]

S. YangX. Kong and C. Tang, A construction of linear codes and their complete weight enumerators, Finite Fields Appl., 48 (2017), 196-226.  doi: 10.1016/j.ffa.2017.08.001.

[22]

J. Yuan and C. Ding, Secret sharing schemes from three classes of linear codes, IEEE Trans. Inf. Theory, 52 (2006), 206-212.  doi: 10.1109/TIT.2005.860412.

[23]

Z. ZhouC. TangX. Li and C. Ding, Binary LCD codes and self-orthogonal codes from a generic construction, IEEE Trans. Inf. Theory, 65 (2019), 16-27.  doi: 10.1109/TIT.2018.2823704.

[24]

W. ZhangH. Yan and H. Wei, Four families of minimal binary linear codes with $w_{min}/w_{max}\leq 1/2$, Appl. Algebra Engrg. Comm. Comput., 30 (2019), 175-184.  doi: 10.1007/s00200-018-0367-x.

Figure 1.  $\mathbb{P} = ([m,n], \leq)$
Table 1.  Theorem 3.1 (Ⅰ)
Weight Frequency
$ 2^{n-1}-i+s(0\le s< i) $ $ 2^{n-i}{i\choose s} $
$ 2^{n-1} $ $ 2^{n-i}-1 $
Weight Frequency
$ 2^{n-1}-i+s(0\le s< i) $ $ 2^{n-i}{i\choose s} $
$ 2^{n-1} $ $ 2^{n-i}-1 $
Table 2.  Theorem 3.1 (Ⅱ)
Weight Frequency
$ 2^{n-1}-(j-m)+t(0\le t< j-m) $ $ 2^{n-(j-m)}{j-m\choose t} $
$ 2^{n-1} $ $ 2^{n-(j-m)}-1 $
Weight Frequency
$ 2^{n-1}-(j-m)+t(0\le t< j-m) $ $ 2^{n-(j-m)}{j-m\choose t} $
$ 2^{n-1} $ $ 2^{n-(j-m)}-1 $
Table 3.  Theorem 3.1 (Ⅲ)
Weight Frequency
$\begin{array}{*{20}{c}} {{2^{n - 1}} + s + t + 2st - (s + 1)(j - m) - (t + 1)i}\\ {0 \le s \le i,0 \le t \le j - m,}\\ {(s,t) \ne (i,j - m)} \end{array}$ $ 2^{n-i-(j-m)}{i\choose s}{j-m\choose t} $
$ 2^{n-1} $ $ 2^{n-i-(j-m)}-1 $
Weight Frequency
$\begin{array}{*{20}{c}} {{2^{n - 1}} + s + t + 2st - (s + 1)(j - m) - (t + 1)i}\\ {0 \le s \le i,0 \le t \le j - m,}\\ {(s,t) \ne (i,j - m)} \end{array}$ $ 2^{n-i-(j-m)}{i\choose s}{j-m\choose t} $
$ 2^{n-1} $ $ 2^{n-i-(j-m)}-1 $
[1]

Irene Márquez-Corbella, Edgar Martínez-Moro, Emilio Suárez-Canedo. On the ideal associated to a linear code. Advances in Mathematics of Communications, 2016, 10 (2) : 229-254. doi: 10.3934/amc.2016003

[2]

Masaaki Harada, Ethan Novak, Vladimir D. Tonchev. The weight distribution of the self-dual $[128,64]$ polarity design code. Advances in Mathematics of Communications, 2016, 10 (3) : 643-648. doi: 10.3934/amc.2016032

[3]

María Chara, Ricardo A. Podestá, Ricardo Toledano. The conorm code of an AG-code. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021018

[4]

Andrew Klapper, Andrew Mertz. The two covering radius of the two error correcting BCH code. Advances in Mathematics of Communications, 2009, 3 (1) : 83-95. doi: 10.3934/amc.2009.3.83

[5]

Denis S. Krotov, Patric R. J.  Östergård, Olli Pottonen. Non-existence of a ternary constant weight $(16,5,15;2048)$ diameter perfect code. Advances in Mathematics of Communications, 2016, 10 (2) : 393-399. doi: 10.3934/amc.2016013

[6]

Laura Luzzi, Ghaya Rekaya-Ben Othman, Jean-Claude Belfiore. Algebraic reduction for the Golden Code. Advances in Mathematics of Communications, 2012, 6 (1) : 1-26. doi: 10.3934/amc.2012.6.1

[7]

Serhii Dyshko. On extendability of additive code isometries. Advances in Mathematics of Communications, 2016, 10 (1) : 45-52. doi: 10.3934/amc.2016.10.45

[8]

Daniel Heinlein, Michael Kiermaier, Sascha Kurz, Alfred Wassermann. A subspace code of size $ \bf{333} $ in the setting of a binary $ \bf{q} $-analog of the Fano plane. Advances in Mathematics of Communications, 2019, 13 (3) : 457-475. doi: 10.3934/amc.2019029

[9]

Andrea Seidl, Stefan Wrzaczek. Opening the source code: The threat of forking. Journal of Dynamics and Games, 2022  doi: 10.3934/jdg.2022010

[10]

Michael Kiermaier, Johannes Zwanzger. A $\mathbb Z$4-linear code of high minimum Lee distance derived from a hyperoval. Advances in Mathematics of Communications, 2011, 5 (2) : 275-286. doi: 10.3934/amc.2011.5.275

[11]

Chengju Li, Sunghan Bae, Shudi Yang. Some two-weight and three-weight linear codes. Advances in Mathematics of Communications, 2019, 13 (1) : 195-211. doi: 10.3934/amc.2019013

[12]

Olof Heden. The partial order of perfect codes associated to a perfect code. Advances in Mathematics of Communications, 2007, 1 (4) : 399-412. doi: 10.3934/amc.2007.1.399

[13]

Sascha Kurz. The $[46, 9, 20]_2$ code is unique. Advances in Mathematics of Communications, 2021, 15 (3) : 415-422. doi: 10.3934/amc.2020074

[14]

Selim Esedoḡlu, Fadil Santosa. Error estimates for a bar code reconstruction method. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1889-1902. doi: 10.3934/dcdsb.2012.17.1889

[15]

Alessandro Barenghi, Jean-François Biasse, Edoardo Persichetti, Paolo Santini. On the computational hardness of the code equivalence problem in cryptography. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2022064

[16]

M. Delgado Pineda, E. A. Galperin, P. Jiménez Guerra. MAPLE code of the cubic algorithm for multiobjective optimization with box constraints. Numerical Algebra, Control and Optimization, 2013, 3 (3) : 407-424. doi: 10.3934/naco.2013.3.407

[17]

Jorge P. Arpasi. On the non-Abelian group code capacity of memoryless channels. Advances in Mathematics of Communications, 2020, 14 (3) : 423-436. doi: 10.3934/amc.2020058

[18]

Masaaki Harada, Takuji Nishimura. An extremal singly even self-dual code of length 88. Advances in Mathematics of Communications, 2007, 1 (2) : 261-267. doi: 10.3934/amc.2007.1.261

[19]

José Gómez-Torrecillas, F. J. Lobillo, Gabriel Navarro. Information--bit error rate and false positives in an MDS code. Advances in Mathematics of Communications, 2015, 9 (2) : 149-168. doi: 10.3934/amc.2015.9.149

[20]

Tonghui Zhang, Hong Lu, Shudi Yang. Two-weight and three-weight linear codes constructed from Weil sums. Mathematical Foundations of Computing, 2022, 5 (2) : 129-144. doi: 10.3934/mfc.2021041

2021 Impact Factor: 1.015

Metrics

  • PDF downloads (84)
  • HTML views (42)
  • Cited by (0)

Other articles
by authors

[Back to Top]