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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.  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. Dougherty, S. 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. 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.  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. 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.  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. 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.  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. Dougherty, S. 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. 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.  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. 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.  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
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