Syndrome decoding of Reed-Muller codes and tensor decomposition over finite fields

Reed-Muller codes are some of the oldest and most widely studied error-correcting codes, of interest for both their algebraic structure as well as their many algorithmic properties. A recent beautiful result of Saptharishi, Shpilka and Volk showed that for binary Reed-Muller codes of length $n$ and distance $d = O(1)$, one can correct $\operatorname{polylog}(n)$ random errors in $\operatorname{poly}(n)$ time (which is well beyond the worst-case error tolerance of $O(1)$). In this paper, we consider the problem of `syndrome decoding' Reed-Muller codes from random errors. More specifically, given the $\operatorname{polylog}(n)$-bit long syndrome vector of a codeword corrupted in $\operatorname{polylog}(n)$ random coordinates, we would like to compute the locations of the codeword corruptions. This problem turns out to be equivalent to a basic question about computing tensor decomposition of random low-rank tensors over finite fields. Our main result is that syndrome decoding of Reed-Muller codes (and the equivalent tensor decomposition problem) can be solved efficiently, i.e., in $\operatorname{polylog}(n)$ time. We give two algorithms for this problem: 1. The first algorithm is a finite field variant of a classical algorithm for tensor decomposition over real numbers due to Jennrich. This also gives an alternate proof for the main result of Saptharishi et al. 2. The second algorithm is obtained by implementing the steps of the Berlekamp-Welch-style decoding algorithm of Saptharishi et al. in sublinear-time. The main new ingredient is an algorithm for solving certain kinds of systems of polynomial equations.Comment: 24 page

Pre-ESRD Depression and Post-ESRD Mortality in Patients with Advanced CKD Transitioning to Dialysis.

Background and objectivesDepression in patients with nondialysis-dependent CKD is often undiagnosed, empirically overlooked, and associated with higher risk of death, progression to ESRD, and hospitalization. However, there is a paucity of evidence on the association between the presence of depression in patients with advanced nondialysis-dependent CKD and post-ESRD mortality, particularly among those in the transition period from late-stage nondialysis-dependent CKD to maintenance dialysis.Design, setting, participants, & measurementsFrom a nation-wide cohort of 45,076 United States veterans who transitioned to ESRD over 4 contemporary years (November of 2007 to September of 2011), we identified 10,454 (23%) patients with a depression diagnosis during the predialysis period. We examined the association of pre-ESRD depression with all-cause mortality after transition to dialysis using Cox proportional hazards models adjusted for sociodemographics, comorbidities, and medications.ResultsPatients were 72±11 years old (mean±SD) and included 95% men, 66% patients with diabetes, and 23% blacks. The crude mortality rate was similar in patients with depression (289/1000 patient-years; 95% confidence interval, 282 to 297) versus patients without depression (286/1000 patient-years; 95% confidence interval, 282 to 290). Compared with patients without depression, patients with depression had a 6% higher all-cause mortality risk in the adjusted model (hazard ratio, 1.06; 95% confidence interval, 1.03 to 1.09). Similar results were found across all selected subgroups as well as in sensitivity analyses using alternate definitions of depression.ConclusionPre-ESRD depression has a weak association with post-ESRD mortality in veterans transitioning to dialysis

A Spectral Bound on Hypergraph Discrepancy

Let $\mathcal{H}$ be a $t$-regular hypergraph on $n$ vertices and $m$ edges. Let $M$ be the $m \times n$ incidence matrix of $\mathcal{H}$ and let us denote $\lambda =\max_{v \perp \overline{1},\|v\| = 1}\|Mv\|$. We show that the discrepancy of $\mathcal{H}$ is $O(\sqrt{t} + \lambda)$. As a corollary, this gives us that for every $t$, the discrepancy of a random $t$-regular hypergraph with $n$ vertices and $m \geq n$ edges is almost surely $O(\sqrt{t})$ as $n$ grows. The proof also gives a polynomial time algorithm that takes a hypergraph as input and outputs a coloring with the above guarantee.Comment: 18 pages. arXiv admin note: substantial text overlap with arXiv:1811.01491, several changes to the presentatio

A Honeycomb Proportional Counter for Photon Multiplicity Measurement in the ALICE Experiment

A honeycomb detector consisting of a matrix of 96 closely packed hexagonal cells, each working as a proportional counter with a wire readout, was fabricated and tested at the CERN PS. The cell depth and the radial dimensions of the cell were small, in the range of 5-10 mm. The appropriate cell design was arrived at using GARFIELD simulations. Two geometries are described illustrating the effect of field shaping. The charged particle detection efficiency and the preshower characteristics have been studied using pion and electron beams. Average charged particle detection efficiency was found to be 98%, which is almost uniform within the cell volume and also within the array. The preshower data show that the transverse size of the shower is in close agreement with the results of simulations for a range of energies and converter thicknesses.Comment: To be published in NIM