On HV-neighborhood group constant sum array

Abstract

A HV-neighborhood group constant sum array with δ distance, is an $m\times n$ array, whose entries are all non-zero elements of an additive Abelian group Γ such that the sum of group elements assigned to the δ-neighborhood of any cell $(i,j)$ in an $m\times n$ array is a unique element $\mu \in \Gamma$, where the δ-neighborhood of a cell $(i,j)$ in an $m\times n$ array is the set of cells that are at most δ distance in the right, left, up, and down from $(i,j)$, excluding the cell $(i,j)$. The element μ is the neighborhood constant sum. In this article, we prove some necessary conditions for the existence of HV-neighborhood group constant sum arrays with δ distance. In addition, if $\delta =1$, a method to construct HV-neighborhood group constant sum arrays and a characterization of HV-neighborhood Klein four-group constant sum arrays are given.