The weight of an $n$-dimensional Boolean vector and addition modulo $2^n$; generalization to the case of modulo $m^n$

Let elements $x$, $y$, $\gamma$ of the residue class $Z_{2^n}$ satisfy the relation $y=x\boxplus \gamma$, where $\boxplus$ is the sign of addition in $Z_{2^n}$. In binary notation the vectors $x$ and $y$ can be regarded as the Boolean vectors $x=(x_1,x_2,\ldots,x_n)$, $y=(y_1,y_2,\ldots,y_n)$ in $B^n$. W e suppose that $x$ is a random element with the uniform distribution on $Z_{2^n}$ and $\gamma$ is a constant. For any $\gamma$ we give the generating function of the two-dimensional distribution of the weights $\xi=|x|$ and $\eta=|y|$, where $|x|=x_1+x_2+\ldots+x_n$, $|y|=y_1+y_2+\ldots+y_n$. The generalization of this result to the case of modulo $m^n$ is also given.
The work was supported by the Russian Foundation for Fundamental Researches, grant 93–011–1443.