Weighted codes in Lee metrics
The authors discuss perfect weighted coverings of radius one with the Manhattan metric in the \(n\)-dimensional infinite grid. This problem can also be stated as a graph coloring problem, as follows. Determine all possible pairs \((a,b)\), such that the \(n\)-dimensional infinite grid can be colored with two colors, where each vertex with the first color is adjacent to \(a\) vertices of the second color and each vertex with the second color is adjacent to \(b\) vertices of the first color. Independently, \textit{D. B. Khoroshilova} [Diskretn. Anal. Issled. Oper. 16, No. 1, 80-92, 97 (2009)] has shown that a sufficient condition for a desired coloring to exist is that \(n \geq (a+b-\gcd(a,b))/2\). In the current work, the constructions are presented in many theorems, so it is difficult to see whether this is actually also the result obtained here; anyway, tabulated results for \(n=9\) hint that this might be the case. Some nonexistence results are also obtained.
scientific article; zbMATH DE number 5614973
scientific article