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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. BritzT. 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. DoughertyS. 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. 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.

[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. 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.

[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. BritzT. 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. DoughertyS. 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. 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.

[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. 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.

[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

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