# American Institute of Mathematical Sciences

• Previous Article
Computing discrete logarithms in cryptographically-interesting characteristic-three finite fields
• AMC Home
• This Issue
• Next Article
Efficient decoding of interleaved subspace and Gabidulin codes beyond their unique decoding radius using Gröbner bases
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. Britz, T. Johnsen and J. Martin, Chains, demi-matroids, and profiles, IEEE Trans. Inform. Theory, 60 (2014), 986-991. doi: 10.1109/TIT.2013.2292524. [2] S. Dodunekov and I. Landgev, On near-MDS codes, Journal of Geometry, 54 (1995), 30-43. doi: 10.1007/BF01222850. [3] S. T. Dougherty, S. Han and H. Liu, Higher weights for codes over rings, Applicable Algebra in Engineering Communication & Computing, 22 (2011), 113-135. [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. [5] G. P. Jian, R. 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. [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. [7] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North Holland, Amsterdam, 1977. [8] C. Martínez-pérez and W. Willems, On the weight hierarchy of product codes, Designs, Codes and Cryptography, 33 (2004), 95-108. [9] B. R. Mcdonald, Linear Algebra Over Commutative Rings (Monographs and Textbooks in Pure and Applied Mathematics, 87), Marcel Dekker, 1984. [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. [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. [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. [13] J. Pieprzyk et al, Ideal Threshold Schemes from MDS Codes, Lecture Notes in Computer Science, Springer Berlin, Heidelberg, 2003. [14] A. Shamir, How to share a secret, Communications of the ACM, 22 (1979), 612-613. doi: 10.1145/359168.359176. [15] V. K. Wei, Generalized Hamming weights for linear codes, IEEE Trans. Inform. Theory, 37 (1991), 1412-1418. doi: 10.1109/18.133259. [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. [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. [18] M. H. Yang, J. Li, K. 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. [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

show all references

##### References:
 [1] T. Britz, T. Johnsen and J. Martin, Chains, demi-matroids, and profiles, IEEE Trans. Inform. Theory, 60 (2014), 986-991. doi: 10.1109/TIT.2013.2292524. [2] S. Dodunekov and I. Landgev, On near-MDS codes, Journal of Geometry, 54 (1995), 30-43. doi: 10.1007/BF01222850. [3] S. T. Dougherty, S. Han and H. Liu, Higher weights for codes over rings, Applicable Algebra in Engineering Communication & Computing, 22 (2011), 113-135. [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. [5] G. P. Jian, R. 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. [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. [7] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North Holland, Amsterdam, 1977. [8] C. Martínez-pérez and W. Willems, On the weight hierarchy of product codes, Designs, Codes and Cryptography, 33 (2004), 95-108. [9] B. R. Mcdonald, Linear Algebra Over Commutative Rings (Monographs and Textbooks in Pure and Applied Mathematics, 87), Marcel Dekker, 1984. [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. [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. [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. [13] J. Pieprzyk et al, Ideal Threshold Schemes from MDS Codes, Lecture Notes in Computer Science, Springer Berlin, Heidelberg, 2003. [14] A. Shamir, How to share a secret, Communications of the ACM, 22 (1979), 612-613. doi: 10.1145/359168.359176. [15] V. K. Wei, Generalized Hamming weights for linear codes, IEEE Trans. Inform. Theory, 37 (1991), 1412-1418. doi: 10.1109/18.133259. [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. [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. [18] M. H. Yang, J. Li, K. 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. [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
 [1] Bagher Bagherpour, Shahrooz Janbaz, Ali Zaghian. Optimal information ratio of secret sharing schemes on Dutch windmill graphs. Advances in Mathematics of Communications, 2019, 13 (1) : 89-99. doi: 10.3934/amc.2019005 [2] Juliang Zhang, Jian Chen. Information sharing in a make-to-stock supply chain. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1169-1189. doi: 10.3934/jimo.2014.10.1169 [3] Stefka Bouyuklieva, Zlatko Varbanov. Some connections between self-dual codes, combinatorial designs and secret sharing schemes. Advances in Mathematics of Communications, 2011, 5 (2) : 191-198. doi: 10.3934/amc.2011.5.191 [4] Ryutaroh Matsumoto. Strongly secure quantum ramp secret sharing constructed from algebraic curves over finite fields. Advances in Mathematics of Communications, 2019, 13 (1) : 1-10. doi: 10.3934/amc.2019001 [5] Sergio R. López-Permouth, Steve Szabo. On the Hamming weight of repeated root cyclic and negacyclic codes over Galois rings. Advances in Mathematics of Communications, 2009, 3 (4) : 409-420. doi: 10.3934/amc.2009.3.409 [6] Nicolas Van Goethem. The Frank tensor as a boundary condition in intrinsic linearized elasticity. Journal of Geometric Mechanics, 2016, 8 (4) : 391-411. doi: 10.3934/jgm.2016013 [7] H. M. Hastings, S. Silberger, M. T. Weiss, Y. Wu. A twisted tensor product on symbolic dynamical systems and the Ashley's problem. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 549-558. doi: 10.3934/dcds.2003.9.549 [8] David Keyes. $\mathbb F_p$-codes, theta functions and the Hamming weight MacWilliams identity. Advances in Mathematics of Communications, 2012, 6 (4) : 401-418. doi: 10.3934/amc.2012.6.401 [9] 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 [10] 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 [11] Alonso sepúlveda Castellanos. Generalized Hamming weights of codes over the $\mathcal{GH}$ curve. Advances in Mathematics of Communications, 2017, 11 (1) : 115-122. doi: 10.3934/amc.2017006 [12] Jinghong Liu, Yinsuo Jia. Gradient superconvergence post-processing of the tensor-product quadratic pentahedral finite element. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 495-504. doi: 10.3934/dcdsb.2015.20.495 [13] Michel H. Geoffroy, Alain Piétrus. Regularity properties of a cubically convergent scheme for generalized equations. Communications on Pure & Applied Analysis, 2007, 6 (4) : 983-996. doi: 10.3934/cpaa.2007.6.983 [14] Fang Chen, Ning Gao, Yao- Lin Jiang. On product-type generalized block AOR method for augmented linear systems. Numerical Algebra, Control & Optimization, 2012, 2 (4) : 797-809. doi: 10.3934/naco.2012.2.797 [15] Teresa Faria, Eduardo Liz, José J. Oliveira, Sergei Trofimchuk. On a generalized Yorke condition for scalar delayed population models. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 481-500. doi: 10.3934/dcds.2005.12.481 [16] Olav Geil, Stefano Martin. Relative generalized Hamming weights of q-ary Reed-Muller codes. Advances in Mathematics of Communications, 2017, 11 (3) : 503-531. doi: 10.3934/amc.2017041 [17] Bin Dan, Huali Gao, Yang Zhang, Ru Liu, Songxuan Ma. Integrated order acceptance and scheduling decision making in product service supply chain with hard time windows constraints. Journal of Industrial & Management Optimization, 2018, 14 (1) : 165-182. doi: 10.3934/jimo.2017041 [18] Weidong Zhao, Yang Li, Guannan Zhang. A generalized $\theta$-scheme for solving backward stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1585-1603. doi: 10.3934/dcdsb.2012.17.1585 [19] Boling Guo, Haiyang Huang. Smooth solution of the generalized system of ferro-magnetic chain. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 729-740. doi: 10.3934/dcds.1999.5.729 [20] Lijun Wei, Xiang Zhang. Limit cycle bifurcations near generalized homoclinic loop in piecewise smooth differential systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2803-2825. doi: 10.3934/dcds.2016.36.2803

2017 Impact Factor: 0.564