Efficiently decoding Reed-Muller codes from random errors

Reed-Muller codes encode an $m$-variate polynomial of degree $r$ by evaluating it on all points in $\{0,1\}^m$. We denote this code by $RM(m,r)$. The minimal distance of $RM(m,r)$ is $2^{m-r}$ and so it cannot correct more than half that number of errors in the worst case. For random errors one may hope for a better result. In this work we give an efficient algorithm (in the block length $n=2^m$) for decoding random errors in Reed-Muller codes far beyond the minimal distance. Specifically, for low rate codes (of degree $r=o(\sqrt{m})$) we can correct a random set of $(1/2-o(1))n$ errors with high probability. For high rate codes (of degree $m-r$ for $r=o(\sqrt{m/\log m})$), we can correct roughly $m^{r/2}$ errors. More generally, for any integer $r$, our algorithm can correct any error pattern in $RM(m,m-(2r+2))$ for which the same erasure pattern can be corrected in $RM(m,m-(r+1))$. The results above are obtained by applying recent results of Abbe, Shpilka and Wigderson (STOC, 2015), Kumar and Pfister (2015) and Kudekar et al. (2015) regarding the ability of Reed-Muller codes to correct random erasures. The algorithm is based on solving a carefully defined set of linear equations and thus it is significantly different than other algorithms for decoding Reed-Muller codes that are based on the recursive structure of the code. It can be seen as a more explicit proof of a result of Abbe et al. that shows a reduction from correcting erasures to correcting errors, and it also bares some similarities with the famous Berlekamp-Welch algorithm for decoding Reed-Solomon codes.Comment: 18 pages, 2 figure

ВОЛЬТАМПЕРОМЕТРИЧЕСКИЙ ЭЛЕКТРОННЫЙ ЯЗЫК С ПРОТОЧНЫМ ТРУБЧАТЫМ ЭЛЕКТРОДОМ ДЛЯ ИДЕНТИФИКАЦИИ СПИРТОСОДЕРЖАЩИХ РАСТВОРОВ

The possibility of on-line voltammetric identification of non-electroactive alcohol solutions by voltammograms of markers reduction on flow-tubular electrode being processed using chemometrics methods has been studied. The voltammetric analyzer IVA-5 and voltammetric cell with tubular work electrode made of stainless steel have been used in the work. Voltammograms of nitrocompounds-markers (o-nitroaniline , o-nitrobenzoic acid, 2,4-dinitrophenol ) have been registered at a constant flow rate of the test solutions through the tubular electrode – 0.6 ml/min; potential scan rate – 0.1 V/s; potential range of the electrode polarization – from 0.0 to -1.0 V. The investigation of multicomponent alcoholic solutions comprising the following steps: three-component mixture of standard markers periodically has been injected into the electrochemical cell and the test solution has been pumped through the tubular electrode. It has been determined that the shape of the voltammograms of nitrocompounds-markers reduction, magnitudes of instantaneous current peaks, their location on the potential axis vary depending on the nature of marker and composition of multicomponent solution under study. Using principal component analysis it has been shown that alcoholic solutions are grouped in clusters on the principal component plane depending on the nature and composition of multicomponent alcohol-containing solution, and the explained variance when using the first three principal components is not less than 84%. Soft independent modeling of class analogies has been shown the possibility of express voltammetric identification of samples with a probability of at least 80% for vodkas and 55% for wines.Keywords: voltammetry, electronic tongue, principal component analysis, tubular electrode, identification, alcohol solutions.(Russian)DOI: http://dx.doi.org/10.15826/analitika.2014.18.2.009 D.M. Bikmeev, A.V. Sidel’nikov, F. Kh. Kudasheva, V.N. MaistrenkoFederal State Budgetary Educational Institution of Higher Professional Education Bashkir State University, Ufa, Bashkortostan, Russian FederationИзучена возможность вольтамперометрической идентификации  спиртосодержащих растворов с проточным трубчатым электродом по вольтамперограммам электровосстановления соединений-маркеров с использованием методов хемометрики. В работе использовали вольтамперометрический анализатор ИВА-5 и вольтамперометрическую ячейку с трубчатым индикаторным электродом из нержавеющей стали. Вольтамперограммы нитросоединений-маркеров (о-нитроанилин, о-нитробензойная кислота, 2,4-динитрофенол) регистрировали при постоянной скорости пропускания исследуемых растворов через трубчатый электрод – 0.6 мл/мин; скорость развертки потенциала – 0.1 В/с; диапазон потенциалов поляризации индикаторного электрода – от 0.0 до -1.0 В. Для идентификации многокомпонентных спиртосодержащих растворов трехкомпонентную смесь нитросоединений-маркеров периодически вводили в электрохимическую ячейку и прокачивали исследуемый раствор через трубчатый электрод. Установлено, что форма вольтамперограмм нитросоединений-маркеров, величины мгновенных токов пиков, их расположение на оси потенциалов изменяются в зависимости от природы маркера и состава спиртосодержащего раствора. С помощью хемометрического метода главных компонент показано, что исследуемые растворы группируются в кластеры на плоскости главных компонент в зависимости от природы и состава спиртосодержащего раствора, а доля объясненной дисперсии при использовании первых трех главных компонент составляет не менее 84 %. Методом формального независимого моделирования аналогий классов показана возможность экспрессной вольтамперометрической идентификации исследуемых образцов с вероятностью не менее 80% для водок и 55 % для вин.Ключевые слова: вольтамперометрия, электронный язык, метод главных компонент, трубчатый электрод, идентификация, спиртосодержащие жидкости.DOI: http://dx.doi.org/10.15826/analitika.2014.18.2.009

Treatment of children with medulloblastoma without metastatic involvement in the age group older than 3 years: international experience and results of intercenter trial

Background. During the past 20 years, some large international studies have been conducted that evaluated the effectiveness of treatment programs for children with medulloblastoma. At the same time, in the standard risk group, fairly high rates of 5-year overall survival (OS) and event-free survival (EFS) were achieved, which amounted to 85% and 80%, respectively. At the present time some risk-adaptive therapeutic programs are developed according to molecular-biological features of tumor cells and possibility of chemotherapy and craniospinal radiation (CSI) therapy dose reduction. Materials and methods. From 2008 to 2018 fifty one pediatric patients with primary diagnosed medulloblastoma in the age group 318 years were included in trial, 38 in standard risk group, 13 in high risk group (without metastatic disease). Treatment program consisted of surgical removal of the primary tumor site with subsequent chemotherapy (with high-dose cyclophosphamide or thiophosphamide) and radiation therapy (with CSI of 23.4 Gy or 36 Gy, depending on the risk group). In order to detect morphological and molecular biological distinctive features of tumor cells, the following criteria were evaluated: histological variant, molecular subgroup, methyltransferase status by DNMT and MGMT proteins expression, presence of C-MYC/N-MYC gene amplification, Iso17q and TP53 gene mutation. Results. As a result of this study, sufficiently high rates of overall survival and progression/relapse-free survival (PRFS) were achieved in standard and high-risk groups patients, which amounted to 76.08.8% and 83.310.8% with median follow-up 62.96.2 months and 52.27.8 months, respectively. There was revealed patients group in the age 37 years with 100% PRFS and median follow-up 66.98.9 months. At the same time, morphological and molecular biological factors of an unfavorable outcome of the disease were absent in the tumor samples (large cell anaplastic histology, C-MYC/N-MYC gene amplification, Iso17q and TP53 gene mutation). We have also achieved 100% PRFS in patients with desmoplastic tumor histology and in patients, who were treated with thiphosphamide-based chemotherapy regimen. Molecular-biological characteristics analysis of tumor cells showed a negative effect on PRFS of DNMT-positive status (Score 4, by 3 markers) and presence of N-MYC gene amplification (SHH molecular subgroup). Conclusion. There was identified a group of patients aged 3 to 7 years, for whom the possibility for reducing of CSR dose down to 18 Gy opens. Understanding of tumor cells methyltransferase status creates the prerequisites for using of epigenetic demethylating therapy. It is necessary more observations to assess the effect of the chemotherapy regimen with high-dose thiophosphamide on the PRFS

Voltammetric electronic tongue with flow-tubular electrode for identification of alcohol solutions

Изучена возможность вольтамперометрической идентификации спиртосодержащих растворов с проточным трубчатым электродом по вольтамперограммам электровосстановления соединений-маркеров с использованием методов хемометрики. В работе использовали вольтамперометрический анализатор ИВА-5 и вольтамперометрическую ячейку с трубчатым индикаторным электродом из нержавеющей стали. Вольтамперограммы нитросоединений-маркеров ( о -нитроанилин, о -нитробензойная кислота, 2,4-динитрофенол) регистрировали при постоянной скорости пропускания исследуемых растворов через трубчатый электрод - 0.6 мл/мин; скорость развертки потенциала - 0.1 В/с; диапазон потенциалов поляризации индикаторного электрода - от 0.0 до -1.0 В. Для идентификации многокомпонентных спиртосодержащих растворов трехкомпонентную смесь нитросоединений-маркеров периодически вводили в электрохимическую ячейку и прокачивали исследуемый раствор через трубчатый электрод. Установлено, что форма вольтамперограмм нитросоединений-маркеров, величины мгновенных токов пиков, их расположение на оси потенциалов изменяются в зависимости от природы маркера и состава спиртосодержащего раствора. С помощью хемометрического метода главных компонент показано, что исследуемые растворы группируются в кластеры на плоскости главных компонент в зависимости от природы и состава спиртосодержащего раствора, а доля объясненной дисперсии при использовании первых трех главных компонент составляет не менее 84 %. Методом формального независимого моделирования аналогий классов показана возможность экспрессной вольтамперометрической идентификации исследуемых образцов с вероятностью не менее 80% для водок и 55 % для вин.The possibility of on-line voltammetric identification of non-electroactive alcohol solutions by voltammograms of markers reduction on flow-tubular electrode being processed using chemometrics methods has been studied. The voltammetric analyzer IVA-5 and voltammetric cell with tubular work electrode made ​​of stainless steel have been used in the work. Voltammograms of nitrocompounds-markers ( o -nitroaniline, o -nitrobenzoic acid, 2,4-dinitrophenol ) have been registered at a constant flow rate of the test solutions through the tubular electrode - 0.6 ml/min; potential scan rate - 0.1 V/s; potential range of the electrode polarization - from 0.0 to -1.0 V. The investigation of multicomponent alcoholic solutions comprising the following steps: three-component mixture of standard markers periodically has been injected into the electrochemical cell and the test solution has been pumped through the tubular electrode. It has been determined that the shape of the voltammograms of nitrocompounds-markers reduction, magnitudes of instantaneous current peaks, their location on the potential axis vary depending on the nature of marker and composition of multicomponent solution under study. Using principal component analysis it has been shown that alcoholic solutions are grouped in clusters on the principal component plane depending on the nature and composition of multicomponent alcohol-containing solution, and the explained variance when using the first three principal components is not less than 84%. Soft independent modeling of class analogies has been shown the possibility of express voltammetric identification of samples with a probability of at least 80% for vodkas and 55% for wines