CONSTRUCTING GRAY MAPS FROM COMBINATORIAL GEOMETRIES

Abstract

The focus of this work is constructing Gray maps for linear codes over a family of Frobenius rings, using tools from combinatorial geometries. The main combinatorial structure that are used are projective geometries P Gn (q), defined over Fq . Codes over Zpk , Galois rings, finite chain rings and Rk have been considered with respect to homogeneous weights. Using hy- perplanes in projective geometries, distance preserving Gray maps from afore- mentioned rings to residue fields have been constructed. The use of projective geometries serves as a novel approach to what was a primarily algebraic ap- proach in the literature. The constructions also serve as a motivation for using homogeneous weights.