A Combinatorial construction of the Gray map over Galois rings

In this correspondence, we will introduce a new combinatorial method for a coordinate-wise construction of the homogeneous-weight preserving Gray map for Galois rings by using elementary tools from Affine Geometries. Our construction differs in the methods used from the algebraic constructions done previously in [M. Greferath, S.E. Schmidt, Gray Isometries for finite chain rings and a nonlinear ternary (36, 3¹², 15) code, IEEE Trans. Inform. Theory 45 (1999) 2522-2524; S. Ling, J.T. Blackford, Z_pk+1-linear codes, IEEE Trans. Inform. Theory 48 (2002) 2592-2605].