Abstract. In this paper we prove new lower bounds for the minimum distance of a toric surface code CP defined by a convex lattice polygon P ⊂ R 2 . The bounds involve a geometric invariant L(P ) , called the full Minkowski length of P which can be easily computed for any given P .

We study the Minkowski length L(P ) of a lattice polytope P , which is defined to be the largest number of non-trivial primitive segments whose Minkowski sum lies in P . The Minkowski length represents the largest possible number of factors in a factorization of polynomials with exponent vectors in P , and shows up in lower bounds for the minimum distance of toric codes. In this paper we give a polytime algorithm for computing L(P ) where P is a 3D lattice polytope.We next study 3D lattice polytopes of Minkowski length 1. In particular, we show that if Q, a subpolytope of P , is the Minkowski sum of L = L(P ) lattice polytopes Q i , each of Minkowski length 1, then the total number of interior lattice points of the polytopes Q 1 , · · · , Q L is at most 4. Both results extend previously known results for lattice polygons. Our methods differ substantially from those used in the two-dimensional case.

This paper is concerned with the minimum distance computation for higher dimensional toric codes defined by lattice polytopes in R n . We show that the minimum distance is multiplicative with respect to taking the product of polytopes, and behaves in a simple way when one builds a k -dilate of a pyramid over a polytope. This allows us to construct a large class of examples of higher dimensional toric codes where we can compute the minimum distance explicitly.

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Let D(m, n) be the set of all the integer points in the mdilate of the Birkhoff polytope of doubly-stochastic n × n matrices. In this paper we find the sharp upper bound on the tropical determinant over the set D(m, n) . We define a version of the tropical determinant where the maximum over all the transversals in a matrix is replaced with the minimum and then find the sharp lower bound on thus defined tropical determinant over D(m, n) .2000 Mathematics Subject Classification. 90C10, 52B12.