  In this work we investigate codes in $\mathbb{Z}_{2^m}^n$ that can correct
errors that occur in just one coordinate of the codeword, with a magnitude of
up to a given parameter $t$. We will show upper bounds on these cross codes,
derive constructions for linear codes and respective decoding algorithm. The
constructions (and decoding algorithms) are given for length $n = 2$ and $n =
3$, but for general $m$ and $t$.
