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In order to determine the Wigner function uniquely, we introduce a newcondition which ensures that the Wigner function has correct marginaldistributions along tilted lines. For a system in $N$ dimensional Hilbertspace, whose "phase space" is a lattice with $N^2$ sites, we get differentresults depending on whether $N$ is odd or even. Under the new condition, theWigner function is determined if $N$ is an odd number, but it does not exist if$N$ is even.Comment: 18 page

Existence of the Wigner function with correct marginal distributions along tilted lines on a lattice