Higher weights and near-MDR codes over chain rings

Zihui Liu; Dajian Liao

doi:10.3934/amc.2018045

Higher weights and near-MDR codes over chain rings

Zihui Liu^*, Dajian Liao

^*Corresponding author for this work

School of Mathematics and Statistics

Jiangsu Ocean University

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

Abstract

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.

Original language	English
Pages (from-to)	761-772
Number of pages	12
Journal	Advances in Mathematics of Communications
Volume	12
Issue number	4
DOIs	https://doi.org/10.3934/amc.2018045
Publication status	Published - Nov 2018

Keywords

Chain condition
Generalized Hamming weight
Near-MDR code
Secret sharing scheme
Tensor product

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10.3934/amc.2018045

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abstract = "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.",

keywords = "Chain condition, Generalized Hamming weight, Near-MDR code, Secret sharing scheme, Tensor product",

author = "Zihui Liu and Dajian Liao",

note = "Publisher Copyright: {\textcopyright} 2018 AIMS.",

year = "2018",

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AB - 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.

KW - Chain condition

KW - Generalized Hamming weight

KW - Near-MDR code

KW - Secret sharing scheme

KW - Tensor product

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Higher weights and near-MDR codes over chain rings

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Keywords

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