On Grid Codes

Abstract

If $A_{i}$ is finite alphabet for $i=1,...,n$, the Manhattan distance is defined in $\prod_{i=1}^{n}A_{i}$. A grid code is introduced as a subset of $\prod_{i=1}^{n}A_{i}$. Alternative versions of the Hamming and Gilbert-Varshamov bounds are presented for grid codes. If $A_{i}$ is a cyclic group for $i=1,...,n$, some bounds for the minimum Manhattan distance of codes that are cyclic subgroups of $\prod_{i=1}^{n}A_{i}$ are determined in terms of their minimum Hamming and Lee distances. Examples illustrating the main results are provided