A construction of $ \mathbb{F}_2 $-linear cyclic, MDS codes

Sara D. Cardell, ,Instituto de Matemática, Estatística e Computação Científica, Universidade Estadual de Campinas, Campinas, Brazil, Joan-Josep Climent, Daniel Panario, Brett Stevens, ,Departament de Matemàtiques, Universitat d'Alacant, Alacant, Spain, ,School of Mathematics and Statistics, Carleton University, Ottawa, Canada
2019 Advances in Mathematics of Communications  
In this paper we construct F 2 -linear codes over F b 2 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.
more » ... 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. 2010 Mathematics Subject Classification: Primary: 94B05, 94B15; Secondary: 94B60.
doi:10.3934/amc.2020047 fatcat:wxpeusoiurbgherekheykualwy