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.

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Advances in Mathematics of Communications
School of Mathematics and Statistics

Cardell, S.D. (Sara D.), Climent, J.-J. (Joan-Josep), Panario, D, & Stevens, B. (2020). A construction of f2-linear cyclic, mds codes. Advances in Mathematics of Communications, 14(3), 437–453. doi:10.3934/amc.2020047