In this work, a flexible and robust private information retrieval (PIR) scheme based on binary non-maximum distance separable (non-MDS) codes is considered. This combines previous works on PIR schemes based on transitive non-MDS codes on one hand, and PIR from MDS-coded Byzantine and non-responsive servers on the other hand. More specifically, a PIR scheme employing binary Reed-Muller (RM) codes tolerant to colluding, Byzantine, and non-responsive servers is constructed, and bounds for the achievable rates are derived under certain conditions. The construction of such schemes turns out to be much more involved than for MDS codes. Namely, the binary query vectors have to be selected with great care to hit the desired information sets, which is technically challenging as will be shown.
This paper considers '$δ$-almost Reed-Muller codes', i.e., linear codes spanned by evaluations of all but a $δ$ fraction of monomials of degree at most $d$. It is shown that for any $δ> 0$ and any $\varepsilon>0$, there exists a family of $δ$-almost Reed-Muller codes of constant rate that correct $1/2-\varepsilon$ fraction of random errors with high probability. For exact Reed-Muller codes, the analogous result is not known and represents a weaker version of the longstanding conjecture that Reed-Muller codes achieve capacity for random errors (Abbe-Shpilka-Wigderson STOC '15). Our approach is …
In this paper we describe a variation of the classical permutation decoding algorithm that can be applied to any affine-invariant code with respect to certain type of information sets. In particular, we can apply it to the family of first-order Reed-Muller codes with respect to the information sets introduced in [2]. Using this algortihm we improve considerably the number of errors we can correct in comparison with the known results in this topic.
Reed-Muller Codes Achieve Capacity on the Binary Erasure Channel under MAP Decoding
我们表明，里德-穆勒码实现容量下的最大后验比特解码传输的二进制擦除信道的所有速率$0 < R < 1$。证明是通用的，并适用于其他代码具有足够的对称性。主要想法是将以下观察结果结合起来：（i）单调函数经历尖锐的阈值行为，（ii）外部信息传递（EXIT）函数是单调的，（iii）Reed-Muller码是2-传递的，因此与它们的码字比特相关联的EXIT函数都是相等的，以及（iv）因此平均EXIT函数的面积定理意味着RM码的阈值是信道容量。
The paper proposes to decode Reed-Muller (RM) codes by projecting onto only a few subspaces such that the number of projections is significantly reduced. It reveals that the probability that error pairs are canceled simultaneously in two different projections is determined by their intersection size. Then, correlation coefficient which indicates the intersection size of two subspaces in a collection is introduced for collecting subspaces. Simulation results show that our proposed approach with a small number of projections onto collected subspaces performs close to the original approach.
We study generalized covering radii, a fundamental property of linear codes that characterizes the trade-off between storage, latency, and access in linear data-query protocols such as PIR. We prove lower and upper bounds on the generalized covering radii of Reed-Muller codes, as well as finding their exact value in certain extreme cases. With the application to linear data-query protocols in mind, we also construct a covering algorithm that gets as input a set of points in space, and find a corresponding set of codewords from the Reed-Muller code that are jointly not farther away from the inp…
The Covering Radius of the Reed--Muller Code $RM(2,7)$ is 40
J. Schatz证明了二阶Reed-Muller码$RM(2, 6)$的覆盖半径为18（IEEE Trans Inf Theory 27：529--530，1985）。然而，$RM(2,7)$的覆盖半径问题多年来一直是一个悬而未决的问题。本文证明了$RM(2,7)$的覆盖半径为40，与$RM(3,7)$中$RM(2,7)$的覆盖半径相同。作为推论，我们还发现了$RM(2,n)$，$n=8,9,10$的新的上界。
Linear codes of 2-designs associated with subcodes of the ternary generalized Reed-Muller codes
本文研究了文献[C。丁角，澳-地Li，Infinite families of 2-designs and 3-designs from linear codes，离散数学340（10）（2017）2415--2431]。给出了这类设计的关联矩阵所生成的三元码的最小距离的一个下界，并证明了这类码是四阶广义Reed-Muller码的子码.
New Results Towards the Characterization of Service Rate Region of Reed-Muller Codes
State Key Laboratory of Cryptography and Digital Economy Security, School of Cyber Science and Technology, Shandong University（山东大学网络空间安全学院密码与数字经济安全全国重点实验室）
