Evaluation Codes from smooth Quadric Surfaces and Twisted Segre Varieties

We give the parameters of any evaluation code on a smooth quadric surface. For hyperbolic quadrics the approach uses elementary results on product codes and the parameters of codes on elliptic quadrics are obtained by detecting a BCH structure of these codes and using the BCH bound. The elliptic quadric is a twist of the surface P^1 x P^1 and we detect a similar BCH structure on twists of the Segre embedding of a product of any d copies of the projective line.Comment: 10 pages. Presented at the conference Workshop on Coding theory and Cryptography 201

Recognizing Treelike k-Dissimilarities

A k-dissimilarity D on a finite set X, |X| >= k, is a map from the set of size k subsets of X to the real numbers. Such maps naturally arise from edge-weighted trees T with leaf-set X: Given a subset Y of X of size k, D(Y) is defined to be the total length of the smallest subtree of T with leaf-set Y . In case k = 2, it is well-known that 2-dissimilarities arising in this way can be characterized by the so-called "4-point condition". However, in case k > 2 Pachter and Speyer recently posed the following question: Given an arbitrary k-dissimilarity, how do we test whether this map comes from a tree? In this paper, we provide an answer to this question, showing that for k >= 3 a k-dissimilarity on a set X arises from a tree if and only if its restriction to every 2k-element subset of X arises from some tree, and that 2k is the least possible subset size to ensure that this is the case. As a corollary, we show that there exists a polynomial-time algorithm to determine when a k-dissimilarity arises from a tree. We also give a 6-point condition for determining when a 3-dissimilarity arises from a tree, that is similar to the aforementioned 4-point condition.Comment: 18 pages, 4 figure

Koszul homology and syzygies of Veronese subalgebras

A graded K-algebra R has property Np if it is generated in degree 1, has relations in degree 2 and the syzygies of order up to p on the relations are linear. The Green\u2013Lazarsfeld index of R is the largest p such that it satisfies the property Np. Our main results assert that (under a mild assumption on the base field) the c-th Veronese subring of a polynomial ring has Green\u2013Lazarsfeld index c + 1. The same conclusion also holds for an arbitrary standard graded algebra, provided C is very large