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Let $F_2$ be the binary field and $Z_{2^r}$ the residue class ring ofintegers modulo $2^r$, where $r$ is a positive integer. For the finite$16$-element commutative local Frobenius non-chain ring $Z_4+uZ_4$, where $u$is nilpotent of index $2$, two weight functions are considered, namely the Leeweight and the homogeneous weight. With the appropriate application of theseweights, isometric maps from $Z_4+uZ_4$ to the binary spaces $F_2^4$ and$F_2^8$, respectively, are established via the composition of otherweight-based isometries. The classical Hamming weight is used on the binaryspace. The resulting isometries are then applied to linear block codes over$Z_4+uZ_4$ whose images are binary codes of predicted length, which may or maynot be linear. Certain lower and upper bounds on the minimum distances of thebinary images are also derived in terms of the parameters of the $Z_4+uZ_4$codes. Several new codes and their images are constructed as illustrativeexamples. An analogous procedure is performed successfully on the ring$Z_8+uZ_8$, where $u^2=0$, which is a commutative local Frobenius non-chainring of order $64$. It turns out that the method is possible in general for theclass of rings $Z_{2^r}+uZ_{2^r}$, where $u^2=0$, for any positive integer $r$,using the generalized Gray map from $Z_{2^r}$ to $F_2^{2^{r-1}}$.

Isometries and Binary Images of Linear Block Codes over $Z_4+uZ_4$ and $Z_8+uZ_8$