• Previous Article
    Efficient decoding of interleaved subspace and Gabidulin codes beyond their unique decoding radius using Gröbner bases
  • AMC Home
  • This Issue
  • Next Article
    Computing discrete logarithms in cryptographically-interesting characteristic-three finite fields
November  2018, 12(4): 761-772. doi: 10.3934/amc.2018045

Higher weights and near-MDR codes over chain rings

1. 

Department of Mathematics, Beijing Institute of Technology, Beijing Key Laboratory on MCAACI, Beijing 100081, China

2. 

College of Science, Huaihai Institute of Technology, Lianyungang 222005, China

* Corresponding author: lzhui@bit.edu.cn

Received  February 2018 Revised  March 2018 Published  September 2018

The matrix description of a near-MDR code is given, and some judging criterions are presented for near-MDR codes. We also give the weight distribution of a near-MDR code and the applications of a near-MDR code to secret sharing schemes. Furthermore, we will introduce the chain condition for free codes over finite chain rings, and then present a formula for computing higher weights of tensor product of free codes satisfying the chain condition. We will also find a chain for any near-MDR code, and thus show that any near-MDR code satisfies the chain condition.

Citation: Zihui Liu, Dajian Liao. Higher weights and near-MDR codes over chain rings. Advances in Mathematics of Communications, 2018, 12 (4) : 761-772. doi: 10.3934/amc.2018045
References:
[1]

T. BritzT. Johnsen and J. Martin, Chains, demi-matroids, and profiles, IEEE Trans. Inform. Theory, 60 (2014), 986-991.  doi: 10.1109/TIT.2013.2292524.  Google Scholar

[2]

S. Dodunekov and I. Landgev, On near-MDS codes, Journal of Geometry, 54 (1995), 30-43.  doi: 10.1007/BF01222850.  Google Scholar

[3]

S. T. DoughertyS. Han and H. Liu, Higher weights for codes over rings, Applicable Algebra in Engineering Communication & Computing, 22 (2011), 113-135.   Google Scholar

[4]

H. Horimoto and K. Shiromoto, On generalized Hamming weights for codes over finite chain rings, Proceedings of the 14th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Melbourne: Lecture Notes in Comput. Sci., 2227 (2001), 141-150.   Google Scholar

[5]

G. P. JianR. Q. Feng and H. F. Wu, Generalized Hamming weights of three classes of linear codes, Finite Fields and Their Applications, 45 (2017), 341-354.  doi: 10.1016/j.ffa.2017.01.001.  Google Scholar

[6]

Z. H. Liu and W. D. Chen, The chain condition of a kind of code of small defects, Mathematics in Practice & Theory (in Chinese), 36 (2006), 314-319.  Google Scholar

[7]

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North Holland, Amsterdam, 1977. Google Scholar

[8]

C. Martínez-pérez and W. Willems, On the weight hierarchy of product codes, Designs, Codes and Cryptography, 33 (2004), 95-108.   Google Scholar

[9]

B. R. Mcdonald, Linear Algebra Over Commutative Rings (Monographs and Textbooks in Pure and Applied Mathematics, 87), Marcel Dekker, 1984.  Google Scholar

[10]

G. H. Norton and A. Sǎlǎgean, On the Hamming distance of linear codes over a finite chain ring, IEEE Trans. Inform. Theory, 46 (2000), 1060-1067.  doi: 10.1109/18.841186.  Google Scholar

[11]

G. H. Norton and A. Sǎlǎgean, On the structure of linear and cyclic codes over a finite chain ring, Applicable Algebra in Engineering Communication & Computing, 10 (2000), 489-506.  doi: 10.1007/PL00012382.  Google Scholar

[12]

M. E. Oued, On MDR codes over a finite ring, International Journal of Information and Coding Theory, 3 (2015), 107-119.  doi: 10.1504/IJICOT.2015.072612.  Google Scholar

[13]

J. Pieprzyk et al, Ideal Threshold Schemes from MDS Codes, Lecture Notes in Computer Science, Springer Berlin, Heidelberg, 2003. Google Scholar

[14]

A. Shamir, How to share a secret, Communications of the ACM, 22 (1979), 612-613.  doi: 10.1145/359168.359176.  Google Scholar

[15]

V. K. Wei, Generalized Hamming weights for linear codes, IEEE Trans. Inform. Theory, 37 (1991), 1412-1418.  doi: 10.1109/18.133259.  Google Scholar

[16]

V. K. Wei and K. Yang, On the generalized Hamming weights of product codes, IEEE Trans. Inform. Theory, 39 (1993), 1709-1713.  doi: 10.1109/18.259662.  Google Scholar

[17]

J. A. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math., 121 (1999), 555-575.  doi: 10.1353/ajm.1999.0024.  Google Scholar

[18]

M. H. YangJ. LiK. Q. Feng and D. D. Lin, Generalized Hamming weights of irreducible cyclic codes, IEEE Trans. Inform. Theory, 61 (2015), 4905-4913.  doi: 10.1109/TIT.2015.2444013.  Google Scholar

[19]

T. S. Zhou, F. Wang, Y. Xin, S. S. Luo, S. H. Qing and Y. X. Yang, A secret sharing scheme based on Near-MDS codes, IEEE International Conference on Network Infrastructure & Digital Content, Beijing, (2009), 833-836 Google Scholar

show all references

References:
[1]

T. BritzT. Johnsen and J. Martin, Chains, demi-matroids, and profiles, IEEE Trans. Inform. Theory, 60 (2014), 986-991.  doi: 10.1109/TIT.2013.2292524.  Google Scholar

[2]

S. Dodunekov and I. Landgev, On near-MDS codes, Journal of Geometry, 54 (1995), 30-43.  doi: 10.1007/BF01222850.  Google Scholar

[3]

S. T. DoughertyS. Han and H. Liu, Higher weights for codes over rings, Applicable Algebra in Engineering Communication & Computing, 22 (2011), 113-135.   Google Scholar

[4]

H. Horimoto and K. Shiromoto, On generalized Hamming weights for codes over finite chain rings, Proceedings of the 14th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Melbourne: Lecture Notes in Comput. Sci., 2227 (2001), 141-150.   Google Scholar

[5]

G. P. JianR. Q. Feng and H. F. Wu, Generalized Hamming weights of three classes of linear codes, Finite Fields and Their Applications, 45 (2017), 341-354.  doi: 10.1016/j.ffa.2017.01.001.  Google Scholar

[6]

Z. H. Liu and W. D. Chen, The chain condition of a kind of code of small defects, Mathematics in Practice & Theory (in Chinese), 36 (2006), 314-319.  Google Scholar

[7]

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North Holland, Amsterdam, 1977. Google Scholar

[8]

C. Martínez-pérez and W. Willems, On the weight hierarchy of product codes, Designs, Codes and Cryptography, 33 (2004), 95-108.   Google Scholar

[9]

B. R. Mcdonald, Linear Algebra Over Commutative Rings (Monographs and Textbooks in Pure and Applied Mathematics, 87), Marcel Dekker, 1984.  Google Scholar

[10]

G. H. Norton and A. Sǎlǎgean, On the Hamming distance of linear codes over a finite chain ring, IEEE Trans. Inform. Theory, 46 (2000), 1060-1067.  doi: 10.1109/18.841186.  Google Scholar

[11]

G. H. Norton and A. Sǎlǎgean, On the structure of linear and cyclic codes over a finite chain ring, Applicable Algebra in Engineering Communication & Computing, 10 (2000), 489-506.  doi: 10.1007/PL00012382.  Google Scholar

[12]

M. E. Oued, On MDR codes over a finite ring, International Journal of Information and Coding Theory, 3 (2015), 107-119.  doi: 10.1504/IJICOT.2015.072612.  Google Scholar

[13]

J. Pieprzyk et al, Ideal Threshold Schemes from MDS Codes, Lecture Notes in Computer Science, Springer Berlin, Heidelberg, 2003. Google Scholar

[14]

A. Shamir, How to share a secret, Communications of the ACM, 22 (1979), 612-613.  doi: 10.1145/359168.359176.  Google Scholar

[15]

V. K. Wei, Generalized Hamming weights for linear codes, IEEE Trans. Inform. Theory, 37 (1991), 1412-1418.  doi: 10.1109/18.133259.  Google Scholar

[16]

V. K. Wei and K. Yang, On the generalized Hamming weights of product codes, IEEE Trans. Inform. Theory, 39 (1993), 1709-1713.  doi: 10.1109/18.259662.  Google Scholar

[17]

J. A. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math., 121 (1999), 555-575.  doi: 10.1353/ajm.1999.0024.  Google Scholar

[18]

M. H. YangJ. LiK. Q. Feng and D. D. Lin, Generalized Hamming weights of irreducible cyclic codes, IEEE Trans. Inform. Theory, 61 (2015), 4905-4913.  doi: 10.1109/TIT.2015.2444013.  Google Scholar

[19]

T. S. Zhou, F. Wang, Y. Xin, S. S. Luo, S. H. Qing and Y. X. Yang, A secret sharing scheme based on Near-MDS codes, IEEE International Conference on Network Infrastructure & Digital Content, Beijing, (2009), 833-836 Google Scholar

[1]

Sushil Kumar Dey, Bibhas C. Giri. Coordination of a sustainable reverse supply chain with revenue sharing contract. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020165

[2]

Thierry Horsin, Mohamed Ali Jendoubi. On the convergence to equilibria of a sequence defined by an implicit scheme. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020465

[3]

Buddhadev Pal, Pankaj Kumar. A family of multiply warped product semi-Riemannian Einstein metrics. Journal of Geometric Mechanics, 2020, 12 (4) : 553-562. doi: 10.3934/jgm.2020017

[4]

Haodong Yu, Jie Sun. Robust stochastic optimization with convex risk measures: A discretized subgradient scheme. Journal of Industrial & Management Optimization, 2021, 17 (1) : 81-99. doi: 10.3934/jimo.2019100

[5]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340

[6]

Susmita Sadhu. Complex oscillatory patterns near singular Hopf bifurcation in a two-timescale ecosystem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020342

[7]

Zonghong Cao, Jie Min. Selection and impact of decision mode of encroachment and retail service in a dual-channel supply chain. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020167

[8]

Anna Abbatiello, Eduard Feireisl, Antoní Novotný. Generalized solutions to models of compressible viscous fluids. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 1-28. doi: 10.3934/dcds.2020345

[9]

Qianqian Han, Xiao-Song Yang. Qualitative analysis of a generalized Nosé-Hoover oscillator. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020346

[10]

Shun Zhang, Jianlin Jiang, Su Zhang, Yibing Lv, Yuzhen Guo. ADMM-type methods for generalized multi-facility Weber problem. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020171

[11]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[12]

Aihua Fan, Jörg Schmeling, Weixiao Shen. $ L^\infty $-estimation of generalized Thue-Morse trigonometric polynomials and ergodic maximization. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 297-327. doi: 10.3934/dcds.2020363

[13]

Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445

[14]

Thomas Frenzel, Matthias Liero. Effective diffusion in thin structures via generalized gradient systems and EDP-convergence. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 395-425. doi: 10.3934/dcdss.2020345

[15]

Jing Zhou, Cheng Lu, Ye Tian, Xiaoying Tang. A socp relaxation based branch-and-bound method for generalized trust-region subproblem. Journal of Industrial & Management Optimization, 2021, 17 (1) : 151-168. doi: 10.3934/jimo.2019104

[16]

Yen-Luan Chen, Chin-Chih Chang, Zhe George Zhang, Xiaofeng Chen. Optimal preventive "maintenance-first or -last" policies with generalized imperfect maintenance models. Journal of Industrial & Management Optimization, 2021, 17 (1) : 501-516. doi: 10.3934/jimo.2020149

2019 Impact Factor: 0.734

Metrics

  • PDF downloads (178)
  • HTML views (318)
  • Cited by (0)

Other articles
by authors

[Back to Top]