Multivariate Vandermonde determinants and toric codes

The minimum distances of toric codes has been studied extensively for various forms of polytopes. In [2], the authors determine bounds for the minimum distance of toric codes for some polytopes P ⊆ R m including the simplices of the form conv(0, `e1, . . . , `en) using Vandermonde determinants. In this paper, we will derive lower and upper bounds to prove the exact minimum distance of some toric codes associated to the special polytopes P = conv(0, `e1, 2`e2, 3`e3) ⊂ R 3 .