Toric surface codes and Minkowski sums

Toric codes are evaluation codes obtained from an integral convex polytope $P \subset \R^n$ and finite field \F_q. They are, in a sense, a natural extension of Reed-Solomon codes, and have been studied recently by J. Hansen and D. Joyner. In this paper, we obtain upper and lower bounds on the minimum distance of a toric code constructed from a polygon $P \subset \R^2$ by examining Minkowski sum decompositions of subpolygons of $P$. Our results give a simple and unifying explanation of bounds of Hansen and empirical results of Joyner; they also apply to previously unknown cases.Comment: 15 pages, 7 figures; This version contains some minor editorial revisions -- to appear SIAM Journal on Discrete Mathematic

Cayley-Bacharach and evaluation codes on complete intersections

In recent work, J. Hansen uses cohomological methods to find a lower bound for the minimum distance of an evaluation code determined by a reduced complete intersection in the projective plane. In this paper, we generalize Hansen's results from P^2 to P^m; we also show that the hypotheses in Hansen's work may be weakened. The proof is succinct and follows by combining the Cayley-Bacharach theorem and bounds on evaluation codes obtained from reduced zero-schemes.Comment: 10 pages. v2: minor expository change

The Evolving Landscape of Internet Control

Over the past two years, we have undertaken several studies at the Berkman Center designed to better understand the control of the Internet in less open societies. During the years we've been engaged in this research, we have seen many incidents that have highlighted the continuing role of the Internet as a battleground for political control, including partial or total Internet shutdowns in China, Iran, Egypt, Libya, and Syria; many hundreds of documented DDoS, hacking, and other cyber attacks against political sites; continued growth in the number of countries that filter the Internet; and dozens of well documented cases of on- and offline persecution of online dissidents. The energy dedicated to these battles for control of the Internet on both the government and dissident sides indicated, if nothing else, that both sides think that the Internet is a critical space for political action. In this paper, we offer an overview of our research in the context of these changes in the methods used to control online speech, and some thoughts on the challenges to online speech in the immediate future