A construction of F2-linear cyclic, MDS codes

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Title: A construction of F2-linear cyclic, MDS codes
Authors: Cardell, Sara D. | Climent, Joan-Josep | Panario, Daniel | Stevens, Brett
Research Group/s: Grupo de Álgebra y Geometría (GAG)
Center, Department or Service: Universidad de Alicante. Departamento de Matemáticas
Keywords: F2-linear code | Cyclicity | MDS codes | Low density parity-check matrix | Index array
Knowledge Area: Álgebra
Issue Date: Aug-2020
Publisher: American Institute of Mathematical Sciences
Citation: Advances in Mathematics of Communications. 2020, 14(3): 437-453. doi:10.3934/amc.2020047
Abstract: In this paper we construct F2-linear codes over Fb2 with length n and dimension n−r where n=rb. These codes have good properties, namely cyclicity, low density parity-check matrices and maximum distance separation in some cases. For the construction, we consider an odd prime p, let n=p−1 and utilize a partition of Zn. Then we apply a Zech logarithm to the elements of these sets and use the results to construct an index array which represents the parity-check matrix of the code. These codes are always cyclic and the density of the parity-check and the generator matrices decreases to 0 as n grows (for a fixed r). When r=2 we prove that these codes are always maximum distance separable. For higher r some of them retain this property.
Sponsor: The first author was supported by CAPES (Brazil). The work of the second author was partially supported by Spanish grants AICO/2017/128 of the Generalitat Valenciana and VIGROB-287 of the Universitat d'Alacant. The third and fourth authors were supported by NSERC (Canada). The first, third and fourth authors acknowledge support from FAPESP SPRINT grant 2016/50476-0.
URI: http://hdl.handle.net/10045/108069
ISSN: 1930-5346 (Print) | 1930-5338 (Online)
DOI: 10.3934/amc.2020047
Language: eng
Type: info:eu-repo/semantics/article
Rights: © 2020 American Institute of Mathematical Sciences
Peer Review: si
Publisher version: https://doi.org/10.3934/amc.2020047
Appears in Collections:INV - GAG - Artículos de Revistas

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