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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 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)
Lifted Reed-Solomon codes are a natural affine-invariant family of error-correcting codes, which generalize Reed-Muller codes. They were known to have efficient local-testing and local-decoding algorithms (comparable with the known algorithms for Reed-Muller codes), but with significantly better rate. We give efficient algorithms for list decoding and local(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 consider the homogeneous components Ur of the map on R = k[x, y, z]/(x A , y B , z C) that multiplies by x + y + z. We prove a relationship between the Smith normal forms of subma-trices of an arbitrary Toeplitz matrix using Schur polynomials, and use this to give a relationship between Smith normal form entries of Ur. We also give a bijective proof of(More)