Isometries and binary images of linear block codes over ℤ4 + uℤ4 and ℤ8 + uℤ8

Abstract

Let be the binary field and ℤ2r the residue class ring of integers modulo 2r, where r is a positive integer. For the finite 16-element commutative local Frobenius non-chain ring ℤ4 + uℤ4, where u is nilpotent of index 2, two weight functions are considered, namely the Lee weight and the homogeneous weight. With the appropriate application of these weights, isometric maps from ℤ4 + uℤ4 to the binary spaces and , respectively, are established via the composition of other weight-based isometries. The classical Hamming weight is used on the binary space. The resulting isometries are then applied to linear block codes over ℤ4+ uℤ4 whose images are binary codes of predicted length, which may or may not be linear. Certain lower and upper bounds on the minimum distances of the binary images are also derived in terms of the parameters of the ℤ4 + uℤ4 codes. Several new codes and their images are constructed as illustrative examples. An analogous procedure is performed successfully on the ring ℤ8 + uℤ8, where u2 = 0, which is a commutative local Frobenius non-chain ring of order 64. It turns out that the method is possible in general for the class of rings ℤ2r + uℤ2r, where u2 = 0, for any positive integer r, using the generalized Gray map from ℤ2r to .