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 .

Abstract. In this paper we consider the following analog of Bezout inequality for mixed volumes:We show that the above inequality is true when ∆ is an n -dimensional simplex and P1, . . . , Pr are convex bodies in R n . We conjecture that if the above inequality is true for all convex bodies P1, . . . , Pr , then ∆ must be an n -dimensional simplex. We prove that if the above inequality is true for all convex bodies P1, . . . , Pr , then ∆ must be indecomposable (i.e. cannot be written as the Minkowski sum of two convex bodies which are not homothetic to ∆ ), which confirms the conjecture when ∆ is a simple polytope and in the 2 -dimensional case. Finally, we connect the inequality to an inequality on the volume of orthogonal projections of convex bodies as well as prove an isomorphic version of the inequality.

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.

A monomial-Cartesian code is an evaluation code defined by evaluating a set of monomials over a Cartesian product. It is a generalization of some families of codes in the literature, for instance toric codes, affine Cartesian codes and J -affine variety codes. In this work we use the vanishing ideal of the Cartesian product to give a description of the dual of a monomial-Cartesian code. Then we use such description of the dual to prove the existence of quantum error correcting codes and MDS quantum error correcting codes. Finally we show that the direct product of monomial-Cartesian codes is a locally recoverable code with t-availability if at least t of the components are locally recoverable codes.

In this paper we give lower bounds for the minimum distance of evaluation codes constructed from complete intersections in toric varieties. This generalizes the results of Gold-Little-Schenck and Ballico-Fontanari who considered evaluation codes on complete intersections in the projective space.Comment: 14 pages, 2 figures. Minor changes, simpler proofs, new example