# American Institute of Mathematical Sciences

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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
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