Bounds on the Minimum Homogeneous Distance of the ${p^{r}}$-ary Image of Linear Block Codes Over the Galois Ring ${\hbox{GR}}(p^{r},m)$

In this correspondence, bounds are derived on the minimum homogeneous distance of the image of a linear block code over the Galois ring ${\hbox{GR}}(p^{r},m)$, with respect to any basis of ${\hbox{GR}}(p^{r},m)$ . These bounds depend on the parameters of ${\hbox{GR}}(p^{r},m)$, the minimum Hamming distance of the block code, and the average value of the homogeneous weight applied on the base ring $ {\BBZ }_{p^{r}}$ . Examples are given of Galois ring codes that meet these bounds.