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 to higher dimensions by requiring their restriction to every subspace of the original dimension to be a codeword of the code being lifted. While the operation is of interest on its own, this work focusses on new ranges of parameters that can be obtained by such codes, in the context of local correction and testing. In particular we present four interesting ranges of parameters that can be achieved by such lifts, all of which are new in the context of affine-invariance and some may be new even in general. The main highlight is a construction of high-rate codes with sublinear time decoding. The only prior construction of such codes is due to Kopparty, Saraf and Yekhanin [33]. All our codes are extremely simple, being just lifts of various parity check codes (codes with one symbol of redundancy), and in the final case, the lift of a Reed-Solomon code.We also present a simple connection between certain lifted codes and lower bounds on the size of "Nikodym sets". Roughly, a Nikodym set in F m q is a set S with the property that every point has a line passing through it which is almost entirely contained in S. While previous lower bounds on Nikodym sets were roughly growing as q m /2 m , we use our lifted codes to prove a lower bound of (1 − o(1))q m for fields of constant characteristic.

BACKGROUND:The Paris System for Reporting Urinary Cytology (TPS) has defined nuclear-to-cytoplasmic (N:C) ratio cutoff values for several of its risk-stratified diagnostic categories. However, because pathologists are not trained to recognize strict N:C ratio cutoff values, a previously designed survey was used to determine whether pathologists could accurately identify N:C ratios according to TPS standards. METHODS: Participants were instructed to estimate the N:C ratio of ideal (line drawing) and real (cell photograph) images presented via an online survey. Actual N:C ratios ranged from 0.3 to 0.8, and 3 answer choices were available: < 0.5, 0.5 and <0.7, and 0.7. The resulting data were analyzed to determine the accuracy and performance of the subgroups. RESULTS: A total of 137 individuals completed the survey.Approximately 24.1% were cytopathologists, 18.2% were pathologists without formal cytopathology training, 18.2% were cytotechnologists, 24.1% were pathology residents, and 15.3% were nonmorphologists. Overall, 70.0%, 67.6%, and 93.3% of responses, respectively, were correct for images with an N:C ratio of < 0.5, 0.5 and < 0.7, and 0.7. For images with an actual N:C ratio < 0.5 and 0.5 and < 0.7, 30.0% and 25.0% of responses, respectively, overestimated the N:C ratio. Furthermore, for images with an N:C ratio of 0.4 and 0.6, > 40.0% of responses overestimated the N:C ratio. As a whole, morphologists were significantly more accurate than nonmorphologists (P 5.030). CONCLUSIONS: Morphologists tended to overestimate the N:C ratio, particularly at ratios close to TPS-recommended cutoff values. Additional training regarding N:C ratio estimation may help pathologists to adapt to this new system.

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 submatrices 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 an identity proven by J. Li and F. Zanello equating the determinant of the middle homogeneous component Ur when (A, B, C) = (a + b, a + c, b + c) to the number of plane partitions in an a × b × c box. Finally, we prove that, for certain vector subspaces of R, similar identities hold relating determinants to symmetry classes of plane partitions, in particular classes 3, 6, and 8.2010 Mathematics Subject Classification. Primary: 05E40; Secondary: 05E05, 05E18, 13E10, 15A15. 2 , b := A−B+C 2 , c := −A+B+C 2 so that A = a + b, B = a + c, C = b + c. Their proof proceeded by evaluating det(U m ) = det(M m (A, B, C)) directly, and comparing the answer to known formulae for such plane partitions. We respond to their call for a more direct, combinatorial explanation (see [10]) with the following:Theorem 1.2. Expressed in the monomial Z-basis for R = Z[x, y, z]/(x a+b , y a+c , z b+c ) the map U m : R m → R m+1 has its determinant det(U m ) equal, up to sign, to its permanent perm(U m ), and each nonzero term in its permanent corresponds naturally to a plane partition in an a × b × c box.

This note corrects Definition 6.3, Proposition 6.7, and Section 7, as specified below. Proposition 6.7 is false for the notion of squarefree game in Definition 6.3, i.e., the equivalent conditions in Proposition 6.2. Henceforth those equivalent conditions define weakly squarefree games. For example, the game on N 3 with rule set {(1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 0)} is weakly squarefree but its P-positions are easily shown not to satisfy Proposition 6.7.The intended notion of squarefree game is any game satisfying the conditions in the following, whose parts mirror Proposition 6.2 as closely as possible.

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 "correction" of the graph, i.e. a graph which has the property and is close to the given graph. For this model, we achieve local property reconstructors for the properties of connectivity and k-connectivity in undirected graphs, and the property of strong connectivity in directed graphs. Along the way, we present a method of transforming a local reconstructor (which acts as a "adjacency matrix oracle" for the corrected graph) into an "adjacency list oracle". This allows us to recursively use our local reconstructor for (k − 1)-connectivity to obtain a local reconstructor for k-connectivity.We also extend this notion of local property reconstruction to parametrized graph properties (for instance, having diameter at most D for some parameter D) and require that the corrected graph has the property with parameter close to the original. We obtain a local reconstructor for the low diameter property, where if the original graph is close to having diameter D, then the corrected graph has diameter roughly 2D.We also exploit a connection between local property reconstruction and property testing, observed by Brakerski, to obtain new tolerant property testers for all of the aforementioned properties. Except for the one for connectivity, these are the first tolerant property testers for these properties.