We propose a new class of efficient decoding algorithms for Reed-Muller (RM)
codes over binary-input memoryless channels. The algorithms are based on
projecting the code on its cosets, recursively decoding the projected codes
(which are lower-order RM codes), and aggregating the reconstructions (e.g.,
using majority votes). We further provide extensions of the algorithms using
list-decoding.
We run our algorithm for AWGN channels and Binary Symmetric Channels at the
short code length ($\le 1024$) regime for a wide range of code rates.
Simulation results show that in both low code rate and high code rate regimes,
the new algorithm outperforms the widely used decoder for polar codes (SCL+CRC)
with the same parameters. The performance of the new algorithm for RM codes in
those regimes is in fact close to that of the maximal likelihood decoder.
Finally, the new decoder naturally allows for parallel implementations.

本文提出了一种基于余类投影的 RM 码高效解码算法，此算法适用于二进制输入无记忆信道；同时提供了基于列表的算法扩展， 并进行了通过模拟实验的性能比较，结果表明本算法在低、高码率下表现优于极化码（SCL + CRC），且在这些范围内的性能接近于最大似然解码器，而且新型解码器支持并行实现。

递归投影聚合解码 Reed-Muller 代码

Recursive projection-aggregation decoding of Reed-Muller codes

Big Data is one of the major challenges of statistical science and has
numerous consequences from algorithmic and theoretical viewpoints. Big Data
always involve massive data but they also often include online data and data
heterogeneity. Recently some statistical methods have been adapted to process
Big Data, like linear regression models, clustering methods and bootstrapping
schemes. Based on decision trees combined with aggregation and bootstrap ideas,
random forests were introduced by Breiman in 2001. They are a powerful
nonparametric statistical method allowing to consider in a single and versatile
framework regression problems, as well as two-class and multi-class
classification problems. Focusing on classification problems, this paper
proposes a selective review of available proposals that deal with scaling
random forests to Big Data problems. These proposals rely on parallel
environments or on online adaptations of random forests. We also describe how
related quantities -- such as out-of-bag error and variable importance -- are
addressed in these methods. Then, we formulate various remarks for random
forests in the Big Data context. Finally, we experiment five variants on two
massive datasets (15 and 120 millions of observations), a simulated one as well
as real world data. One variant relies on subsampling while three others are
related to parallel implementations of random forests and involve either
various adaptations of bootstrap to Big Data or to "divide-and-conquer"
approaches. The fifth variant relates on online learning of random forests.
These numerical experiments lead to highlight the relative performance of the
different variants, as well as some of their limitations.

本文主要关注于基于分类问题的随机森林模型处理大数据问题的现有方案，并尝试了五个变体实验，探究它们的相对性能和局限性。