Reed-Muller codes for random erasures and errors

This paper studies the parameters for which Reed-Muller (RM) codes over $GF(2)$ can correct random erasures and random errors with high probability, and in particular when can they achieve capacity for these two classical channels. Necessarily, the paper also studies properties of evaluations of multi-variate $GF(2)$ polynomials on random sets of inputs. For erasures, we prove that RM codes achieve capacity both for very high rate and very low rate regimes. For errors, we prove that RM codes achieve capacity for very low rate regimes, and for very high rates, we show that they can uniquely decode at about square root of the number of errors at capacity. The proofs of these four results are based on different techniques, which we find interesting in their own right. In particular, we study the following questions about $E(m,r)$, the matrix whose rows are truth tables of all monomials of degree $\leq r$ in $m$ variables. What is the most (resp. least) number of random columns in $E(m,r)$ that define a submatrix having full column rank (resp. full row rank) with high probability? We obtain tight bounds for very small (resp. very large) degrees $r$, which we use to show that RM codes achieve capacity for erasures in these regimes. Our decoding from random errors follows from the following novel reduction. For every linear code $C$ of sufficiently high rate we construct a new code $C'$, also of very high rate, such that for every subset $S$ of coordinates, if $C$ can recover from erasures in $S$, then $C'$ can recover from errors in $S$. Specializing this to RM codes and using our results for erasures imply our result on unique decoding of RM codes at high rate. Finally, two of our capacity achieving results require tight bounds on the weight distribution of RM codes. We obtain such bounds extending the recent \cite{KLP} bounds from constant degree to linear degree polynomials

Optimal and Efficient Decoding of Concatenated Quantum Block Codes

We consider the problem of optimally decoding a quantum error correction code -- that is to find the optimal recovery procedure given the outcomes of partial "check" measurements on the system. In general, this problem is NP-hard. However, we demonstrate that for concatenated block codes, the optimal decoding can be efficiently computed using a message passing algorithm. We compare the performance of the message passing algorithm to that of the widespread blockwise hard decoding technique. Our Monte Carlo results using the 5 qubit and Steane's code on a depolarizing channel demonstrate significant advantages of the message passing algorithms in two respects. 1) Optimal decoding increases by as much as 94% the error threshold below which the error correction procedure can be used to reliably send information over a noisy channel. 2) For noise levels below these thresholds, the probability of error after optimal decoding is suppressed at a significantly higher rate, leading to a substantial reduction of the error correction overhead.Comment: Published versio

Hidden Markov Models and their Application for Predicting Failure Events

We show how Markov mixed membership models (MMMM) can be used to predict the degradation of assets. We model the degradation path of individual assets, to predict overall failure rates. Instead of a separate distribution for each hidden state, we use hierarchical mixtures of distributions in the exponential family. In our approach the observation distribution of the states is a finite mixture distribution of a small set of (simpler) distributions shared across all states. Using tied-mixture observation distributions offers several advantages. The mixtures act as a regularization for typically very sparse problems, and they reduce the computational effort for the learning algorithm since there are fewer distributions to be found. Using shared mixtures enables sharing of statistical strength between the Markov states and thus transfer learning. We determine for individual assets the trade-off between the risk of failure and extended operating hours by combining a MMMM with a partially observable Markov decision process (POMDP) to dynamically optimize the policy for when and how to maintain the asset.Comment: Will be published in the proceedings of ICCS 2020; @Booklet{EasyChair:3183, author = {Paul Hofmann and Zaid Tashman}, title = {Hidden Markov Models and their Application for Predicting Failure Events}, howpublished = {EasyChair Preprint no. 3183}, year = {EasyChair, 2020}

FGF21 mediates the thermogenic and insulin-sensitizing effects of dietary methionine restriction but not its effects on hepatic lipid metabolism

© 2017 by the American Diabetes Association. Dietary methionine restriction (MR) produces a rapid and persistent remodeling of white adipose tissue (WAT), an increase in energy expenditure (EE), and enhancement of insulin sensitivity. Recent work established that hepatic expression of FGF21 is robustly increased by MR. Fgf212/2 mice were used to test whether FGF21 is an essential mediator of the physiological effects of dietary MR. The MR-induced increase in energy intake and EE and activation of thermogenesis in WAT and brown adipose tissue were lost in Fgf212/2 mice. However, dietary MR produced a comparable reduction in body weight and adiposity in both genotypes because of a negative effect of MR on energy intake in Fgf212/2 mice. Despite the similar loss in weight, dietary MR produced a more significant increase in in vivo insulin sensitivity in wild-Type than in Fgf212/2 mice, particularly in heart and inguinal WAT. In contrast, the ability of MR to regulate lipogenic and integrated stress response genes in liver was not compromised in Fgf212/2 mice. Collectively, these findings illustrate that FGF21 is a critical mediator of the effects of dietary MR on EE, remodeling of WAT, and increased insulin sensitivity but not of its effects on hepatic gene expression

Characterization of a Chromosomally Encoded 2,4-Dichlorophenoxyacetic Acid/a-Ketoglutarate Dioxygenase from \u3ci\u3eBurkholderia\u3c/i\u3e sp. Strain RASC

The findings of previous studies indicate that the genes required for metabolism of the pesticide 2,4-dichlorophenoxyacetic acid (2,4-D) are typically encoded on broad-host-range plasmids. However, characterization of plasmid-cured strains of Burkholderia sp. strain RASC, as well as mutants obtained by transposon mutagenesis, suggested that the 2,4-D catabolic genes were located on the chromosome of this strain. Mutants of Burkholderia strain RASC unable to degrade 2,4-D (2,4-D- strains) were obtained by insertional inactivation with Tn5. One such mutant (d1) was shown to have Tn5 inserted in tfdARASC, which encodes 2,4-D/alpha-ketoglutarate dioxygenase. This is the first reported example of a chromosomally encoded tfdA. The tfdARASC gene was cloned from a library of wild-type Burkholderia strain RASC DNA and shown to express 2,4-D/alpha-ketoglutarate dioxygenase activity in Escherichia coli. The DNA sequence of the gene was determined and shown to be similar, although not identical, to those of isofunctional genes from other bacteria. Moreover, the gene product (TfdARASC) was purified and shown to be similar in molecular weight, amino-terminal sequence, and reaction mechanism to the canonical TfdA of Alcaligenes eutrophus JMP134. The data presented here indicate that tfdA genes can be found on the chromosome of some bacterial species and suggest that these catabolic genes are rather mobile and may be transferred by means other than conjugation

Good Quantum Convolutional Error Correction Codes And Their Decoding Algorithm Exist

Quantum convolutional code was introduced recently as an alternative way to protect vital quantum information. To complete the analysis of quantum convolutional code, I report a way to decode certain quantum convolutional codes based on the classical Viterbi decoding algorithm. This decoding algorithm is optimal for a memoryless channel. I also report three simple criteria to test if decoding errors in a quantum convolutional code will terminate after a finite number of decoding steps whenever the Hilbert space dimension of each quantum register is a prime power. Finally, I show that certain quantum convolutional codes are in fact stabilizer codes. And hence, these quantum stabilizer convolutional codes have fault-tolerant implementations.Comment: Minor changes, to appear in PR

Quantifying the Performance of Quantum Codes

We study the properties of error correcting codes for noise models in the presence of asymmetries and/or correlations by means of the entanglement fidelity and the code entropy. First, we consider a dephasing Markovian memory channel and characterize the performance of both a repetition code and an error avoiding code in terms of the entanglement fidelity. We also consider the concatenation of such codes and show that it is especially advantageous in the regime of partial correlations. Finally, we characterize the effectiveness of the codes and their concatenation by means of the code entropy and find, in particular, that the effort required for recovering such codes decreases when the error probability decreases and the memory parameter increases. Second, we consider both symmetric and asymmetric depolarizing noisy quantum memory channels and perform quantum error correction via the five qubit stabilizer code. We characterize this code by means of the entanglement fidelity and the code entropy as function of the asymmetric error probabilities and the degree of memory. Specifically, we uncover that while the asymmetry in the depolarizing errors does not affect the entanglement fidelity of the five qubit code, it becomes a relevant feature when the code entropy is used as a performance quantifier.Comment: 21 pages, 10 figure

Mixed quantum state detection with inconclusive results

We consider the problem of designing an optimal quantum detector with a fixed rate of inconclusive results that maximizes the probability of correct detection, when distinguishing between a collection of mixed quantum states. We develop a sufficient condition for the scaled inverse measurement to maximize the probability of correct detection for the case in which the rate of inconclusive results exceeds a certain threshold. Using this condition we derive the optimal measurement for linearly independent pure-state sets, and for mixed-state sets with a broad class of symmetries. Specifically, we consider geometrically uniform (GU) state sets and compound geometrically uniform (CGU) state sets with generators that satisfy a certain constraint. We then show that the optimal measurements corresponding to GU and CGU state sets with arbitrary generators are also GU and CGU respectively, with generators that can be computed very efficiently in polynomial time within any desired accuracy by solving a semidefinite programming problem.Comment: Submitted to Phys. Rev.

An iterative algorithm for parametrization of shortest length shift registers over finite rings

The construction of shortest feedback shift registers for a finite sequence S_1,...,S_N is considered over the finite ring Z_{p^r}. A novel algorithm is presented that yields a parametrization of all shortest feedback shift registers for the sequence of numbers S_1,...,S_N, thus solving an open problem in the literature. The algorithm iteratively processes each number, starting with S_1, and constructs at each step a particular type of minimal Gr\"obner basis. The construction involves a simple update rule at each step which leads to computational efficiency. It is shown that the algorithm simultaneously computes a similar parametrization for the reciprocal sequence S_N,...,S_1.Comment: Submitte

Multi-Exciton Spectroscopy of a Single Self Assembled Quantum Dot

We apply low temperature confocal optical microscopy to spatially resolve, and spectroscopically study a single self assembled quantum dot. By comparing the emission spectra obtained at various excitation levels to a theoretical many body model, we show that: Single exciton radiative recombination is very weak. Sharp spectral lines are due to optical transitions between confined multiexcitonic states among which excitons thermalize within their lifetime. Once these few states are fully occupied, broad bands appear due to transitions between states which contain continuum electrons.Comment: 12 pages, 4 figures, submitted for publication on Jan,28 199