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

Abstract

Let F2 be the binary field and Z2r the residue class ring of integers modulo 2 r , where r is a positive integer. For the finite 16-element commutative local Frobenius nonchain 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 F 4 2 and F 8 2 , 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 u 2 = 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 u 2 = 0, for any positive integer r, using the generalized Gray map from Z2r to F 2 r−1 2 . Index Terms-Frobenius ring, homogeneous weight, linear block code, binary image.