Learn More
In this work we explore error-correcting codes derived from the "lifting" of "affine-invariant" codes. Affine-invariant codes are simply linear codes whose coordinates are a vector space over a field and which are invariant under affine-transformations of the coordinate space. Lifting takes codes defined over a vector space of small dimension and lifts them(More)
The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Abstract We prove NP-hardness results for five of Nintendo's largest video game franchises: Mario, all Legend of Zelda games; all Metroid games; and all Pokémon role-playing games. In addition, we prove PSPACE-completeness of the Donkey(More)
The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Abstract A local property reconstructor for a graph property is an algorithm which, given oracle access to the adjacency list of a graph that is " close " to having the property, provides oracle access to the adjacency matrix of a "(More)
A non-crossing connected graph is a connected graph on vertices arranged in a circle such that its edges do not cross. The count for such graphs can be made naturally into a q-binomial generating function. We prove that this generating function exhibits the cyclic sieving phenomenon, as conjectured by S.-P. Eu. Résumé. Un graphe connexe dont les sommets(More)
  • Alan Guo
  • 2013
We present a general framework for constructing high-rate error correcting codes that are locally correctable (and hence locally decodable if linear) with a sublinear number of queries, based on lifting codes with respect to functions on the coordinates. Our approach generalizes the lifting of affine-invariant codes (of Guo, Kopparty, and Sudan) and its(More)
We provide a winning strategy for sums of games of Mark-t, an impartial game played on the nonnegative integers where each move consists of subtraction by an integer between 1 and t − 1 inclusive, or division by t, rounding down when necessary. Our algorithm computes the Sprague-Grundy values for arbitrary n in quadratic time. This addresses one of the(More)
We encode arbitrary finite impartial combinatorial games in terms of lattice points in rational convex polyhedra. Encodings provided by these lattice games can be made particularly efficient for octal games, which we generalize to squarefree games. These additionally encompass all heap games in a natural setting, in which the Sprague–Grundy theorem for(More)
We prove that the class of locally testable affine-invariant properties is closed under sums, intersections and " lifts ". The sum and intersection are two natural operations on linear spaces of functions, where the sum of two properties is simply their sum as a vector space. The " lift " is a less natural property that has been studied before. Previously(More)
We investigate the minimum distance of the error correcting code formed by the homomorphisms between two finite groups G and H. We prove some general structural results on how the distance behaves with respect to natural group operations , such as passing to subgroups and quotients, and taking products. Our main result is a general formula for the distance(More)